diff --git "a/1NFST4oBgHgl3EQfWTh0/content/tmp_files/2301.13780v1.pdf.txt" "b/1NFST4oBgHgl3EQfWTh0/content/tmp_files/2301.13780v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/1NFST4oBgHgl3EQfWTh0/content/tmp_files/2301.13780v1.pdf.txt" @@ -0,0 +1,2151 @@ +Commuting Cohesions +David Jaz Myers +Mitchell Riley +February 1, 2023 +Abstract +Shulman’s spatial type theory internalizes the modalities of Lawvere’s axiomatic cohesion in a homotopy +type theory, enabling many of the constructions from Schreiber’s modal approach to differential cohomology to +be carried out synthetically. In spatial type theory, every type carries a spatial cohesion among its points and +every function is continuous with respect to this. But in mathematical practice, objects may be spatial in more +than one way at the same time; a simplicial space has both topological and simplicial structures. Moreover, +many of the constructions of Schreiber’s differential cohomology and Schreiber and Sati’s account of proper +equivariant orbifold cohomology require the interplay of multiple sorts of spatiality — differential, equivariant, +and simplicial. +In this paper, we put forward a type theory with “commuting focuses” which allows for types to carry +multiple kinds of spatial structure. The theory is a relatively painless extension of spatial type theory, and +enables us to give a synthetic account of simplicial, differential, equivariant, and other cohesions carried by the +same types. We demonstrate the theory by showing that the homotopy type of any differential stack may be +computed from a discrete simplicial set derived from the ˇCech nerve of any good cover. We also give other +examples of multiple cohesions, such as differential equivariant types and supergeometric types, laying the +groundwork for a synthetic account of Schreiber and Sati’s proper orbifold cohomology. +Contents +1 +Introduction +2 +2 +A Type Theory with Commuting Focuses +6 +3 +Specializing a Focus +11 +3.1 +Detecting Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +3.2 +Detecting Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +4 +Examples of Focuses +14 +4.1 +Real Cohesions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.2 +Simplicial Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.2.1 +The ˇCech Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +4.3 +Global Equivariant Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +4.4 +Topological Toposes +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +5 +Multiple Focuses +25 +5.1 +Commuting Cohesions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +6 +Examples with Multiple Focuses +29 +6.1 +Simplicial Real Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +6.2 +Equivariant Differential Cohesion +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +6.3 +Supergeometric Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +1 +arXiv:2301.13780v1 [math.CT] 31 Jan 2023 + +A Proof Sketches for Admissible Rules +36 +1 +Introduction +Homotopy type theory is a novel foundation of mathematics which centers the notion of identification of mathe- +matical objects. In homotopy type theory, every mathematical object is of a certain type of mathematical object; +and, if x and y are both objects of type X, then we know by virtue of the definition of the type X what it means +to identify x with y as elements of the type X. For example, if x and y were real vector spaces (so that X was the +type of real vector spaces), then to identify x with y would be to give a R-linear isomorphism between them. If x +and y were smooth manifolds, then to identify them would be to give a diffeomorphism between them. If x and +y were mere numbers, then to identify them would be simply to prove them equal. And so on, for any type of +mathematical object. +Homotopy theory, in the abstract, is the study of the identifications of mathematical objects. Homotopy type +theory is well suited for synthetic homotopy theory (e.g. [12, 24, 13, 18] and many others), but to apply these +theorems in algebraic topology — where objects are identified by giving continuous deformations of one into the +other — requires a modification to the theory. To emphasize the difference here, compare the higher inductive +circle S1, which is the type freely generated by a point with a self-identification, with the topological circle S1 +defined as the set of points in the real plane with unit distance from the origin: +S1 ≡ {(x,y) : R2 | x2 +y2 = 1}. +The base point of the higher inductive circle S1 has many non-trivial self-identifications, whereas two points of the +topological circle may be identified (in a unique way) just when they are equal. The two types are closely related +however: the higher inductive circle S1 is the homotopy type of the topological circle S1 obtained by identifying +the points of the latter by continuous deformation. Ordinary homotopy type theory does not have the language to +express this relationship, and therefore cannot apply the synthetic theorems concerning the higher inductive circle +to topological questions about the topological circle. +What is needed is a way to distinguish between types which carry topological structure and discrete types +with only homotopical structure. In his Cantor’s ‘Lauter Einsen’ and Cohesive Toposes [27], Lawvere points out +that this distinction between natively cohesive and discrete sets is already present in the writings of Cantor as the +distinction between the Menge of mathematical practice and the abstract Kardinalzahlen which arise by abstracting +away from the relationships among the points of a space. In the paper, and his subsequent Axiomatic Cohesion +[28], Lawvere formalizes this opposition between cohesion and discreteness as an adjoint triple between toposes: +Mengen +Kardinalen +points +codiscrete +discrete +This adjoint triple induces an adjoint pair of idempotent (co)monads on the topos of spaces or Mengen: the +left adjoint, ♭, retopologizes a space with the discrete topology, and the right adjoint, ♯, retopologizes it with the +codiscrete topology. Lawvere notes that in many cases — when the spaces in question are “locally connected” — +there will be a fourth adjoint π0 on the left which produces the discrete set of connected components of a space; +this system of adjoint functors characterizes his axiomatic cohesion. +But the real power of Lawvere’s axiomatic cohesion is unlocked by Schreiber’s move from 1-toposes whose +objects are cohesive sets to ∞-toposes whose objects are cohesive homotopy types. In his Differential Cohomology +in a Cohesive ∞-Topos (DCCT) [44], Schreiber shows that Lawvere’s axiomatics, when interpreted in ∞-toposes, +give rise to the hexagonal fracture diagrams which characterize differential cohomology — alongside many other +observations about the centrality of the defining adjoints of cohesion in higher topology and physics. What was +the functor π0 that took the connected components of a space becomes, in the ∞-categorical setting, the functor +2 + +Π∞ which takes the shape (in the sense of Lurie [31, §7.1.6]) of a stack. All in all, a cohesive ∞-logos has three +adjoint endofunctors +S ⊣ ♭ ⊣ ♯ +where S takes the shape or homotopy type of a higher space considered as a discrete space, ♭ takes its under- +lying homotopy type of discrete points, and ♯ takes the underlying homotopy type of points but retopologized +codiscretely. +In Brouwer’s Fixed Point Theorem in Real-Cohesive Homotopy Type Theory [47] (henceforth Real Cohesion), +Shulman brings this distinction between cohesive Mengen and discrete Karndinalen to homotopy type theory via +his spatial type theory. Spatial type theory internalizes the ♭ and ♯ modalities from Schreiber’s DCCT which relate +discrete (but homotopically interesting) types like S1 and spatial types like S1. Spatial type theory also improves +upon a previous axiomatization of these modalities in HoTT due to Schreiber and Shulman [45], by replacing +axioms with judgemental rules. Cohesive homotopy type theory is spatial type theory with an additional axiom that +implies the local contractibility of the sorts of spaces in question; from this axiom the further left adjoint S to ♭ may +be defined. +Homotopy type theory may be interpreted into any ∞-topos [26, 46], so that a type in homotopy type theory +becomes a sheaf of homotopy types externally. In particular, if we interpret the topological circle S1 defined as +a subset of R2 into the ∞-topos of sheaves on the site of continuous manifolds, it becomes the sheaf (of sets) +represented by the external continuous manifold S1, while the higher inductive circle S1 gets interpreted as the +constant sheaf at the homotopy type of the circle. By the Yoneda lemma, then, any function definable on S1 is +necessarily continuous. Since functions f : X → Y in HoTT are defined simply by specifying an element f(x) : Y +in the context of a free variable x : X, variation in a free variable confers a liminal sort of continuity: such an +expression could be interpreted in a spatial ∞-topos in which case it necessarily defines a continuous function. +Shulman’s spatial type theory works by introducing the notion of a crisp free variable to get around this liminal +continuity. An expression in spatial type theory depends on its crisp free variables discontinuously. The modalities +♭ and ♯ of spatial type theory represent crisp variables universally on the left and right respectively. In this way, ♭X +is the discrete retopologization of the spatial type X, while ♯X is its codiscrete retopologization — a map out of ♭X +is a discontinuous map out of X, while a map into ♯X is a discontinuous map into X. +Spatial type theory is intended to be interpreted into local geometric morphisms γ : E → S of ∞-toposes, +those for which γ∗ has a fully faithful right adjoint γ! which gives a geometric morphism f : S → E (with f∗ := ��!) +adjoint to γ which acts as the focal point of E as a space over S . The adjoint modalities ♭ and ♯ are interpreted +as the adjoint idempotent (co)monad pair γ∗γ∗ and γ!γ∗ respectively. A crisp free variable is then one which varies +over an object of the focal point S : a free variable is crisp when it is in focus. +There is not only one way for mathematical objects to be spatial. Spaces may cohere with smooth, analytic, +algebraic, condensed, and simplicial or cubical combinatorial structures — and more. Each of these cases would +give rise to a particular spatial type theory as the internal language of an appropriate local ∞-topos. But there are +many cases arising in practice where we need not just one axis of spatiality, but many at once. For example, it is +a classical theorem that the homotopy type of a manifold may be computed as the realization of a (topologically +discrete) simplicial set associated to the ˇCech nerve of a good open cover of the manifold. This theorem relates a +simplicial set to a continuous space, via an intermediary simplicial space which is both continuous and simplicial +at the same time — the ˇCech nerve of the cover. But in spatial homotopy type theory there is only one notion of +crisp variable, and therefore just one sort of spatiality. +For simplicial types, the discrete reflection is the 0-skeleton sk0, while the codiscrete reflection is the 0- +coskeleton. For simplicial spaces, we then have both the (topologically) discrete ♭ and codiscrete ♯, as well as +the simplicially 0-skeletal sk0 and 0-coskeletal csk0. Interestingly, the ˇCech nerve itself arises from these modal- +ities: the ˇCech nerve of a map f : X → Y between 0-skeletal types (that is, continuous or differential stacks with +no simplicial structure) is its csk0-image, as we will see later in Proposition 4.2.13. A simplicial space has both a +shape SX and a realization (or colimit) reX; the first is a topologically discrete simplicial type, while the latter is a +0-skeletal but spatial type. With all these modalities, we can prove the theorem about good covers described above +as Theorem 6.1.5. +3 + +Another use case for multiple axes of spatiality is Sati and Schreiber’s Proper orbifold cohomology [43], where +orbifolds are understood both as having both differential structure (as differential stacks) and global equivariant +structure (concerning their singularities). In order to get the correct generalized cohomology of orbifolds without +relying on ad-hoc constructions based on a global quotient presentation of the orbifold, Sati and Schreiber work +with the ∞-topos of global equivariant differential stacks, which is local both over the global equivariant topos and +the topos of differential stacks. Here the differential modalities S, ♭ and ♯ are augmented with the modalities of +equivariant cohesion [41]: +< +, +⊂ +, and +≺ +, which take the strict quotient, the underlying space as an invariant type, +and the Yoneda embedding of the underlying space of a global equivariant type respectively. Again the modalities +play a central role in the theory, with the ordinary Borel cohomology of a global quotient orbifold X �G being the +ordinary cohomology of S +⊂ +(X �G), while the proper equivariant Bredon cohomology of X �G is the cohomology +of +≺ +(X �G), twisted by the map to +≺ +BG classifying the quotient map +≺ +X → +≺ +(X �G). +In these cases, modalities that lie in the same position in their adjoint chain commute with each other, so, for +example, ♭ commutes with sk0 and ♯ commutes with csk0. However, there are cases where these modalities are +nested, with one spatiality being a refinement of another. This occurs for example in supergeometry as formulated +by Schreiber in [44] with the modalities of solid cohesion. The supergeometric focus is given by the even comodal- +ity ⇒ (which takes the even part of a superspace) and the rheonomic modality Rh which is given by localizing at +the odd line R0|1. +In this paper, we put forward a modification of spatial type theory to allow for multiple axes of spatiality. +Our theory works by allowing for a meet semi-lattice of focuses ♥,♣,..., each with a separate notion of ♥-crisp +variable and pair of adjoint (co)modalities ♭♥ and ♯♥. Like spatial type theory, our custom type theory gets us to +the coalface of synthetic homotopy theory very efficiently while staying simple enough to be used in an informal +style. +The presence of multiple notions of crispness forces a more complex context structure than spatial type theory’s +separation of the context into a crisp zone and cohesive zone. Similar to many other modal type theories [30, 22, +21, 9, 38], we annotate each variable with modal information, here, the focuses for which that variable is crisp. The +typing rules for the modalities of each focus then work essentially independently. The exception is ♭-elimination, +which is upgraded to allow the crispness of the term being eliminated to be maintained in the variable bound by +the induction (a ‘crisp’ induction principle). +Ours is far from the only extension of type theory with multiple modalities, but as we discuss in more detail +later, no existing theory has the combination of features that we are looking for: dependent types (ruling out [30]) +that may depend on modal variables (ruling out [9]), multiple commuting comodalities (ruling out [47, 11, 38]) +each with a with right-adjoint modality (ruling out [33]) and no further left-adjoints (ruling out [22, 21] and [16, +§14]). +In addition to allowing us to formalize the theorem about ˇCech nerves of open covers as Theorem 6.1.5, our +type theory will be able to handle the equivariant differential cohesion used by Sati and Schreiber in their Proper +orbifold cohomology [43], as well as the nested focuses of Schreiber’s supergeometric solid cohesion [44]. This +extends the work of Cherubini [17] and the first author [34, 35, 36] of giving synthetic accounts of the constructions +of Schreiber [44] and Sati-Schreiber [43]. +Positing an additional focus does not disturb arguments made using existing focuses, so we also expect our +theory to be helpful when dipping into simplicial arguments in the course of other reasoning by adding a simplicial +focus and making use of the new modalities. The problem of defining simplicial types in ordinary Book HoTT +remains open, and there are now a number of different approaches to constructing simplicial types which each use +some extension to the underlying type theory. In this paper, we will axiomatize the 1-simplex ∆[1] as a linear order +with distinct top and bottom and use the cohesive modalities to define the ˇCech nerve of a map and the realization +or colimit of a simplicial type. We believe our approach here would pair nicely with other approaches to simplicial +types for the purposes of synthetic (∞,1)-category theory such as [42, 14, 49, 48], where the sk0 modality would +take the core of a Rezk type.1 +1Though we have not looked in detail at how the focuses would work with the Riehl-Shulman simplicial type theory, and in particular +how they would interact with the cubes/topes zones of the Riehl-Shulman context. +4 + +Outline of the present paper. +After presenting our type theory in §2, we will look at ways to specialize the +spatiality of a focus in §3. In particular, we will observe that in many cases there is a small class of test spaces +Gi so that codiscreteness (that is, being ♯-modal) is detected by uniquely lifting against the ♭-counits ♭Gi → Gi; +such Gi will be said to detect continuity. Externally, the Gi could be any family which generates the logos under +colimits. In practice, the Gi will be test spaces which minimally carry the appropriate spatiality: in the simplicial +case, the simplices ∆[n]; in the real-cohesive case, the Euclidean spaces Rn; for condensed sets, the profinite sets, +etc. +In §3, we will also meet a family of axioms which hold for spatialities that are locally contractible. For +example, continuous manifolds which are built from Euclidean spaces by colimits are locally contractible, while +condensed sets which are built from profinite sets by colimits need not be locally contractible. In general, a space +is locally contractible when it has a constant shape in the sense of Lurie [31, §7.1.6]. +We may define a space C to be contractible when any map C → S to a discrete space S is constant. If the +converse holds — a space S is discrete (♭-modal) if every map C → S is constant — then we say that C detects the +connectivity of spaces. For example, R detects the connectivity of continuous ∞-groupoids, and ∆[1] detects the +connectivity of simplicial ∞-groupoids. If there is a space (or family of spaces) which detects connectivity, then +the local geometric morphism p corresponding to the morphism is furthermore strongly locally contractible in that +p∗ has a left adjoint p! which takes the (constant value of the) shape of a space. In the case that p is both local and +strongly locally contractible, we say that p is cohesive following Lawvere [28], Schreiber [44], and Shulman [47]. +Nullifying at the family of spaces which detect connectivity gives a modality S which is left adjoint to ♭; it may be +thought of as taking the homotopy type of a space. +In §4 we will give example axioms for specializing single focuses. We will review Shulman’s axioms for +real cohesion, where the Euclidean spaces Rn detect continuity and connectivity. We will then see simplicial +cohesion in some detail, where the simplices ∆[n] detect continuity and connectivity. We give our types simplicial +structure by axiomatizing the 1-simplex ∆[1] as a total order with distinct top and bottom elements, following +Joyal’s characterization of simplicial sets as the classifying topos for such orders [50]. We use the csk0 modality +to construct ˇCech nerves of maps. Then we will describe the global equivariant cohesion first observed by Rezk +[41] and used by Sati and Schreiber in [43]. Finally, we will briefly describe axioms for topological focuses such +as Johnstone’s topological topos of sequential spaces [25] and the condensed/pyknotic topos of Clausen-Scholze +[19] and Barwick-Haine [10]. +After surveying some of the different sorts of spatiality which types might carry, we turn our attention to +multiple focuses in §5. In Definition 5.1.3, we define what it means for two cohesions to be orthogonal: when the +family which detects the connectivity of one is discrete with respect to the other, and vice versa. We then prove a +few lemmas concerning orthogonal cohesions, in particular concerning when it is possible to commute the various +modalities past each other. +Finally, we give examples of multiple focuses in §6. We begin with simplicial real cohesion, which has both +a simplicial focus and a real-cohesive focus which are orthogonal. We prove, in Theorem 6.1.5, that the shape of +any 0-skeletal type M may be computed as the realization of a topologically discrete simplicial type constructed +from the ˇCech nerve of any good cover U of M — one for which finite intersections of the Ui are contractible in +the sense of being S-connected. +Next, we combine equivariant cohesion with differential cohesion to give the series of modalities used in Sati +and Schreiber’s Proper orbifold cohomology [43]. Happily, no extra axioms are needed to show that the two +cohesions are orthogonal; we prove this in Lemma 6.2.1. +Finally, we describe the supergeometric or “solid” cohesion of Schreiber’s Differential Cohomology in a Co- +hesive ∞-topos. This extends real cohesion with the odd line R0|1, where the “discrete” comodality of the super- +geometric focus takes the even part of a supergeometric space, and the “codiscrete” modality takes a rheonomic +reflection of the space, one whose super structure is uniquely determined by its even structure. Unlike our other +examples where the focuses involved are orthogonal, here the differential focus is included in the supergeometric +focus: any discrete space is also purely even, as is any codiscrete space. +Acknowledgements. We would like to thank Urs Schreiber for his careful reading and extensive comments during +5 + +the drafting process of the paper. And we would like to thank Hisham Sati for his feedback and words of encour- +agement. The authors are grateful for the support of Tamkeen under the NYU Abu Dhabi Research Institute grant +CG008. +2 +A Type Theory with Commuting Focuses +The fundamental duality in higher topos theory is between the ∞-topos — a general sort of space — and the ∞- +logos — the category of sheaves of homotopy types on such a space [7]. This duality is perfect: a map of ∞-toposes +E → F is defined to be a lex accessible functor Sh∞(F) → Sh∞(E ) between their corresponding ∞-logoses in the +opposite direction. +This duality between toposes and logoses gives a nice perspective on the distinction between the petite toposes, +which are used as generalized spaces in practice, and the gros toposes — or rather, their dual logoses — which +are used as categories of spaces, rather than as spaces in their own right. Quite opposite to their names, the +petite toposes are “big” spaces, while the gros toposes are “small” spaces; it is their dual logoses which are +correctly described by the adjectives “petite” and “gros”. Since the logos is the category of sheaves on the topos, +or equivalently the category of ´etale maps into the topos, the “larger” the topos the more constraining the ´etale +condition becomes. For that reason, the gros toposes have qualitatively “smaller” categories of sheaves. On the +other hand, the more general the ´etale spaces may be, the “smaller” the base topos must be. In general, the “biggest” +logoses, the logoses of spaces, must correspond to the “smallest” toposes: those toposes which are infinitesimal +patches around a focal point. This point of view is emphasized in Chapter 4 of DCCT [44]. +We may therefore, as a first pass, identify logoses of spaces as dual to those toposes E which are local over a +focal point F. A geometric morphism p : E → F is local when it admits a left adjoint right inverse f : F → E +in the (∞,2)-category of toposes which we call the focal point of p. If E is a topological space (that is, if its +corresponding logos Sh∞(E ) is the category of sheaves Sh∞(X) on a sober topological space X), then the terminal +geometric morphism γ : E → S is local just when X has a focal point: a point f ∈ X whose only open neighborhood +is the whole of X. In particular, the prime spectrum of a ring A is local if and only if A is a local ring; in this case, +the focal point is the unique maximal ideal. +On the logos side, this means that the direct image p∗ admits a fully faithful right adjoint p! (which is f∗). All +together, this gives an adjoint triple between the corresponding logoses: +Sh∞(E ) +Sh∞(F) +p∗ +p! +p∗ +Thinking of the objects of Sh∞(E ) as generalized spaces and the objects of Sh∞(F) as mere homotopy types +(sheaves on a point), we may see the direct image p∗ as taking the underlying homotopy type of points of a space, +while p∗ and p! are the discrete and codiscrete topologizations of bare homotopy types, respectively. This adjoint +triple gives rise to an adjoint pair +p∗p∗ ⊣ p!p∗ +of a idempotent comonad p∗p∗ and idempotent monad p!p∗ on the logos Sh∞(E ). Understood as operations on +spaces, these are the discrete and codiscrete retopologizations of a space respectively. +Examples of local toposes with focal point F having category of sheaves Sh∞(F) = ∞Grpd the ∞-category +of ∞-groupoids include simplicial types ∞Grpd∆op (where discrete is 0-skeletal and codiscrete is 0-coskeletal), +continuous and differentiable ∞-groupoids2 Sh∞({Rn}) (where discrete means all charts are constant, and codis- +crete means that any function valued in the set of points is a chart), condensed ∞-groupoids (where discrete means +discrete, and codiscrete means codiscrete), and global equivariant ∞-groupoids ∞GrpdGloop (where discrete means +2These are the gros toposes of C 0 and C ∞ manifolds, respectively. +6 + +being a constant presheaf on the global orbit category, and codiscrete means being a presheaf representable by an +ordinary ∞-groupoid). +Shulman [46] has shown that every ∞-logos may be presented by a model of homotopy type theory, allowing +reasoning conducted in homotopy type theory to be interpreted in any ∞-logos. In this sense, homotopy type theory +is to ∞-logoses as set theory is to the 1-logoses of Grothendieck, Lawvere, and Tierney. In Brouwer’s Fixed Point +Theorem in Real-Cohesive Homotopy Type Theory [47], Shulman also put forward a spatial type theory which +may (conjecturally) be interpreted into any local geometric morphism. Spatial type theory is characterized by +including an adjoint pair ♭ ⊣ ♯ of a lex comodality ♭ and lex modality ♯. These are to be interpreted as p∗p∗ and +p!p∗ respectively. +In spatial type theory, any type has a spatial structure. The existence of this spatial structure is witnessed by +the two opposite ways that we can get rid of it: either we can remove all the spatial relationships between points, +using the “discrete” ♭ comodality, or we can trivialize the spatial relations using the “codiscrete” ♯ modality. We +emphasize that this spatial structure is distinct from the homotopical structure that all types have by virtue of the +identifications between their elements. For example, the topological circle +S1 := {(x,y) : R2 | x2 +y2 = 1} +has a spatial structure as a subset of the Euclidean plane (as a sheaf on the site of continuous manifolds, for +example), but is a homotopy 0-type (or “set”) without any non-trivial identifications between its points; in particular +ΩS1 = ∗. The homotopy type S1 of the circle, however, is spatially discrete but has many non-trivial identifications +of its point: in particular ΩS1 = Z. +There is not, however, only one way to be spatial in mathematics. For example a simplicial topological space +has both a simplicial structure and a topological structure. This can be witnessed at the level of toposes as well. If +p : E → F admits a focal point f : F → E , then f ∆op : F ∆op → E ∆op is also a focal point of p∆op : E ∆op → F ∆op, +where the logos Sh∞(E ∆op) := (Sh∞(E ))∆op is the category of simplicial objects in the logos Sh∞(E ). But there is +another local geometric morphism γ : E ∆op → E where γ∗ sends a simplicial sheaf X• to X0 and γ! is given by the +0-coskeleton csk0 Sn := Sn. These two different axes of spatiality on the objects of Sh∞(E )∆op commute, in that the +following diagram of adjoints commutes: +Sh∞(E )∆op +Sh∞(F)∆op +Sh∞(E ) +Sh∞(F) +p∆op +∗ +γ∗ +γ∗ +p∗ +In particular, we have that p∗∆op p∗∆op and γ∗γ∗ commute as endofunctors of Sh∞(E )∆op. The former discretely +retopologizes a simplicial space, while the latter includes the space of 0-simplices as a 0-skeletal simplicial space. +Each focus gives an axis along which the objects of the top logos Sh∞(E )∆op may carry spatial structure. +When working in, say, simplicial differential spaces, we would like to have access to both the S ⊣ ♭ ⊣ ♯ of real +cohesion and the re ⊣ sk0 ⊣ csk0 of simplicial cohesion. Shulman’s spatial type theory offers no way to do this: the +♭ and sk0 comonads have incompatible claims on the notion of ‘crisp’ variable. +The solution is to allow a separate notion of crispness for each focus we are interested in. In this section, we +will describe the rules for a type theory with commuting focuses, generalizing ordinary spatial type theory in the +case of a single non-trivial focus. We will then describe axioms which make these into commuting cohesions, in +the sense of cohesive type theory. +To this end, we will fix an commutative idempotent monoid Focus of focuses; we will write the product of the +focus ♥ and the focus ♣ as ♥♣. This product induces an ordering on the focuses by saying that ♣ ≤ ♥ whenever +7 + +♣♥ = ♣. With respect to this ordering, the product becomes the meet; we may therefore also think of Focus as a +meet semi-lattice. We will write the identity element of Focus as ⊤, and note that it is the top focus with respect to +the order. +For most of our purposes in this paper, our commutative idempotent monoid Focus of focuses will be freely +generated by a finite set of basic focuses. Explicitly, we may take Focus = Pf (BasicFocuses)op to be the set of +finite subsets of the set of basic focuses with union as the product, and therefore the opposite of the ordering of +subsets by inclusion. +All variables in the context will be annotated with the focus that they are in: +x :♥ X ⊢ t : T +In general, we will abbreviate the context entry x :⊤ X as x : X. In the case that Focus = {♥ ≤ ⊤} is freely +generated by one basic focus, we recover the split context used in Shulman’s spatial type theory, where our context +x :♥ X, y :⊤ Y ctx corresponds to Shulman’s context x :: X | y : Y ctx. +To describe the typing rules, we will need a couple of auxiliary operations on contexts. The first operation ♥Γ +adds a specific focus ♥ to the annotation on every variable in a context. So: +♥(·) :≡ · +♥(Γ,x :♣ A) :≡ (♥Γ),x :♥♣ A +We also need an operation ♥ \ Γ that deletes any variables not contained within a given focus ♥; this is the +equivalent of going from ∆ | Γ ctx to ∆ | · ctx in ordinary spatial type theory. +♥\(·) :≡ · +♥\(Γ,x :♣ A) :≡ +� +(♥\Γ),x :♣ A +if ♣ ≤ ♥ +♥\Γ +otherwise +We say that a variable x :♣ X is ♥-crisp if ♣ ≤ ♥, and so the ♥-crisp variables are precisely those that survive +the ♥\Γ operation. We say that a term t : T is ♥-crisp if both it and its type T only contain ♥-crisp variables, i.e., +it is well-formed in context ♥\Γ. +We are now ready to describe the rules of the type theory. All the usual type formers — Σs, Πs, etc. — will be +included as usual, only referring to variables of the top focus ⊤. By the convention that x :⊤ X be written as x : X, +these rules look exactly as they do usually. We therefore focus on the new features of type theory with commuting +focuses. +Structural Rules. +CTX-EMPTY · ctx +CTX-EXT ♥\Γ ⊢ A type +Γ,x :♥ A ctx +VAR +Γ,x :♥ A,Γ′ ctx +Γ,x :♥ A,Γ′ ⊢ x : A +In prose, these rules read as follows: +• CTX-EMPTY: The empty context is a context. +• CTX-EXT: If A is a ♥-crisp type in context Γ, then Γ,x :♥ A ctx is a context. +• VAR: If x :♥ A appears in a context, then the variable x has type A in that context. +Remark 2.0.1. Given a context Γ,x :♥ A ctx, it must be the case that A only depends on the variables in Γ which +are themselves ♥-crisp. This careful context formation rule is what replaces the division of the context into two +zones in Shulman’s spatial type theory. In the conclusion of the variable rule, the type A is well-formed in context +Γ,x :♥ A,Γ′ ctx by the admissible DIVIDE-WK rule given below, followed by further weakening with Γ′. +8 + +Remark 2.0.2. Rather than annotating variables, may be tempting to try a floating context separator |♥ for each +focus, so that the variables to the left of |♥ are precisely the ♥-crisp ones. Such contexts are not sufficiently +general; specifically, the ♭-elimination rule will let us produce a context containing x :♥ A,y :♣ B which clearly +cannot be separated in this way. +The following rules and equations will be made admissible, with the proofs sketched in Appendix A. +WK +Γ,Γ′ ⊢J +Γ,x :♥ A,Γ′ ⊢J +−−−−−−−−−− +SUBST +♥\Γ ⊢ a : A +Γ,x :♥ A,Γ′ ⊢J +Γ,Γ′[a/x] ⊢J [a/x] +−−−−−−−−−−−−−−−−−−−−− +PROMOTE-CTX +Γ ctx +♥Γ ctx +−−−− +PROMOTE +Γ ⊢J +♥Γ ⊢J +−−−−− +DIVIDE-CTX +Γ ctx +♥\Γ ctx +−−−−−− +DIVIDE-WK +♥\Γ ⊢J +Γ ⊢J +−−−−−− +♥(♣Γ) ≡ (♥♣)Γ +♥\(♣\Γ) ≡ (♣♥)\Γ +• First, we have ordinary weakening by a variable, and a ‘crisp’ substitution similar to that used in spatial +type theory, where crisp variables may only be substituted with similarly crisp terms. These specialize to the +ordinary weakening and substitution rules when used for ♥ ≡ ⊤. +• PROMOTE-CTX corresponds to the application of the endofunctor ♥ to the context Γ, and PROMOTE to +precomposition with the counit morphism ♥Γ → Γ. +• DIVIDE-CTX gives the largest ‘subcontext’ ♥ \ Γ of Γ such that there is a substitution Γ → ♥(♥ \ Γ). The +context operation ♥ \ − thus acts like a left-adjoint to ♥−, although semantically a left-adjoint may not +exist. +Rules for ♭. +We now come to the rules for the ♭ comodality. +♭-FORM ♥\Γ ⊢ A type +Γ ⊢ ♭♥A type +♭-INTRO ♥\Γ ⊢ M : A +Γ ⊢ M♭♥ : ♭♥A +♭-ELIM +♣♥\Γ ⊢ A type +Γ,x :♣ ♭♥A ⊢ C type +♣\Γ ⊢ M : ♭♥A +Γ,u :♣♥ A ⊢ N : C[u♭♥/x] +Γ ⊢ (let u♭♥ := M inN) : C[M/x] +♭-BETA +♣♥\Γ ⊢ A type +Γ,x :♣ ♭♥A ⊢ C type +♣♥\Γ ⊢ K : A +Γ,u :♣♥ A ⊢ N : C[u♭♥/x] +Γ ⊢ (let u♭♥ := K♭♥ inN) ≡ N[K/u] : C[K♭♥/x] +In prose, these rules read as follows: +• ♭-FORM: If A is a ♥-crisp type, then we may form ♭♥A type. +• ♭-INTRO: If M is a ♥-crisp term of type A, then we may form M♭♥ of type ♭♥A. +• ♭-ELIM: If C is a type depending on the ♣-crisp variable x :♣ ♭♥A, and M : ♭♥A is a ♣-crisp element of +type ♭♥A, then we may assume that M is of the form u♭♥ for a ♣♥-crisp variable u :♣♥ A when defining an +9 + +element of C[M/x]. We write this element as (let u♭♥ := M inN) : C[M/x] where N : C[u♭♥/x] is the element +we defined assuming that M was of the form u♭♥. The equation ♥\(♣\Γ) ≡ (♣♥)\Γ is necessary here to +know that the type ♭♥A is well-formed in context ♣\Γ. +• ♭-BETA: If M actually is of the form K♭♥ for suitably crisp K, then we simply substitute K in for u. The term +K must be ♣♥-crisp for both the ♭-INTRO and ♭-ELIM to have been applied, and so its substitution for the +♣♥-crisp variable u is well-formed. +Remark 2.0.3. These rules are stronger than the ones used by Shulman for spatial type theory, even in the case of +a single focus. We have built in a ♣-crisp induction principle for ♭♥, for any two focuses ♥ and ♣: if the term we +are inducting on is already ♣-crisp, then we may maintain that crispness in the new assumption u. +If we have a single non-trivial focus ♥, as is the case in Shulman’s type theory, then taking ♣ = ♥ in the above +expression yields the ‘crisp ♭ induction’ principle of [47, Lemma 5.1]. This induction principle is proven by taking +a detour through ♯, but here we choose to build it into the rule directly. +Our elimination rule is in fact also admissible from the less general one that requires the freshly bound variable +to only be ♥-crisp, but we choose the more general rule for convenience. +Rules for ♯. +The rules for ♯ are a little simpler, and in the case of a single focus specialize exactly to the rules of +spatial type theory. +♯-FORM ♥Γ ⊢ A type +Γ ⊢ ♯♥A type +♯-INTRO +♥Γ ⊢ M : A +Γ ⊢ M♯♥ : ♯♥A +♯-ELIM ♥\Γ ⊢ N : ♯♥A +Γ ⊢ N♯♥ : A +♯-BETA +♥\Γ ⊢ M : A +Γ ⊢ (M♯♥)♯♥ ≡ M : A +♯-ETA +Γ ⊢ N : ♯♥A +Γ ⊢ N ≡ (N♯♥)♯♥ : ♯♥A +In prose, these rules read as follows: +• ♯-FORM: When forming the type ♯♥A, all variables may be used in A as though they are ♥-crisp. +• ♯-INTRO: When forming a term M♯♥ : ♯♥A, all variables may be used in M as though they are ♥-crisp. +• ♯-ELIM: If N is a ♥-crisp element of ♯♥A, we may extract an element N♯♥ : A. +• ♯-BETA: If M is a ♥-crisp element of A, then M♯♥♯♥ ≡ M. +• ♯-ETA: Any term of N : ♯♥A is definitionally equal to N♯♥ +♯♥. As in ordinary spatial type theory, the term N♯♥ +may not be well-typed on its own, because it may use non-crisp variables of the context Γ. It is however +well-typed underneath the outer (−)♯♥, since the introduction rule allows us to use any variable as though it +is ♥-crisp. +Remark 2.0.4. Perhaps surprisingly, the shape of the ♯-FORM and ♯-INTRO rules is what builds the left-exactness +of ♭ into the theory. This is the case even in ordinary spatial type theory, not a feature that only appears in this +multi-focus setting. The trick is that the promotion operation ♥Γ distributes over the context extensions in Γ rather +than being a ‘stuck’ context former applied to Γ as a whole. Specifically, when using ♯ to derive crisp Id-induction, +one applies ♯ to a type +x :: A,y :: A, p :: (x = y) | · ⊢ C type, +yielding a type +· | x : A,y : A, p : (x = y) ⊢ ♯C type. +Internalized, the former context represents the type (x : ♭A)×(y : ♭A)×♭(x♭ = y♭), but ♯-FORM treats it as identical +to ♭((x : A)×(y : A)×(x = y)) when applying adjointness. +10 + +Remark 2.0.5. In most cases of interest, our commutative idempotent monoid of focuses is freely generated by +a finite set of basic focuses. In this situation, it suffices to provide the ♭ and ♯ only for the basic focuses, as the +remainder can be derived. The top focus ⊤ (which semantically corresponds to the entire topos we are working in) +has both ♭⊤A and ♯⊤A canonically equivalent to A. And given focuses ♥ and ♣, it is quickly proven that ♭♥♣ is +equivalent to ♭♥♭♣ and similarly for the ♯s. +Related Type Theories. +Besides the original spatial type theory, there are several other dependent modal type +theories that come close to our needs. +The ‘adjoint type theory’ perspective [40, 29, 30] was the guiding principle that led to the original spatial type +theory of [47]. Indeed, when instantiated with appropriate mode theory, the framework of [30] reproduces a simply +typed version of the theory presented here. The specific mode theory to be used is a cartesian monoid with a system +of commuting, product-preserving endomorphisms. A dependently typed variant of adjoint type theory is not yet +forthcoming, but we expect that our dependent type theory would be an instance of it. +An separate line of work on modal type theories is Multimodal Type Theory [22, 23]. In MTT, every mode +morphism µ is reified in the type theory as a positive type former, and each modality modµ must have a left- +adjoint-like context operator written �µ. If we do not assume the existence of S, then we are only able to describe +♯ in this way. +Later work [21] describes a multimodal type theory where each mode morphism becomes a (more convenient) +negative type former. The semantic requirements are even stronger: the functor corresponding to the modality +must be a dependent right-adjoint [11], whose left adjoint is itself a parametric right adjoint. This is too strict even +to capture ♯ without additional assumptions. +In [16, §14], an alternative ‘cohesive type theory’ is presented, using a combination of the above two styles +of modal operator. Rather than working with the endofunctors on the topos of interest, the cohesive setting is +kept as an adjoint quadruple Π0 ⊣ Disc ⊣ Γ ⊣ CoDisc. A positive type former is used for Disc and negative type +formers for Γ and CoDisc, due to the requirements on having one or two left-adjoints. It is likely that this could +be extended to commuting cohesions, but the interactions of the various context �− operations for the left-adjoints +may be difficult to describe. +The type theory with context structure most formally similar to ours is ParamDTT [38, 37], where variables +in the context annotated with a modality indicate a variable under that modality directly, not its left adjoint. It +is from this work that we take the left-division notation − \ Γ for the clearing operation on contexts, which itself +has appeared in other guises, for example [39, 2, 3]. ParamDTT uses a fixed ‘mode theory’ with three modalities +{¶,id,♯} equipped with a particular composition law, but it is clear that the rules for contexts and basic type formers +would work equally well for other sets of modalities. A version of the cohesive ♭ can be derived from the ‘modal +Σ-type’, fixing the second component to be the unit type. There does not appear to be a way to derive the ordinary +(negative) rules for ♯ in ParamDTT. +3 +Specializing a Focus +A focus gives a specific axis along which a type may be spatial. In simplicial cohesion, we have a simplicial focus +sk0 ⊣ csk0 and in differential cohesion a differential focus ♭ ⊣ ♯. But what makes the simplicial focus simplicial and +the differential focus differential? In this section, we will investigate two axioms schemes which can determine the +peculiarities of a given focus. In the next section, we will see these axioms in use. +First, we note that with a single focus, type theory with commuting focuses is the same theory as Shulman’s +spatial type theory in [47]. +Theorem 3.0.1. Any of the lemmas and theorems proven in §3, 4, 5, and 6 of Real Cohesion [47] concerning ♭ +and ♯ and using no axioms are true also of ♭♥ and ♯♥ for any fixed focus ♥. Theorems which do involve the use of +axioms are also valid, so long as the crispness used in those axioms is interpreted as ♥-crispness. +Proof. The rules for ♭♥ and ♯♥ specialize to Shulman’s rules, and therefore his proofs carry through directly. +11 + +Specifically, ♭♥ is a coreflector and ♯♥ is a monadic modality, both are lex, and ♭♥ is (♥-crisply) left-adjoint to +♯♥. +Since adding a focus only expands the rules of the type theory and does not restrict the application of any of +the rules for any of the other focuses, any of the theorems proven in this section for a single focus will apply when +working with multiple focuses as well. +For the rest of this section, we will work within a single focus ♥, and for that reason we will drop the anno- +tations by ♥ in our expressions. For example, we will write ♭♥ as simply ♭, and we will write x :♥ X as x :: X, +following Shulman. +3.1 +Detecting Continuity +In this section, we will look at an axiom which ties the liminal sort of “continuity” implied by the crisp variables +of the type theory to the concrete continuity of a particular type G. +Our axiom will take the form of a lifting property characterizing those crisp maps which are ♯-modal. As we +will show in the upcoming Theorem 3.1.2, a crisp map is ♯-modal if and only if it lifts crisply (in a sense made +precise in Definition 3.1.1) against all of the ♭-counits. +Definition 3.1.1. Let c :: A → B and f :: X → Y be crisp maps. We say that c lifts crisply against f if for any crisp +square as on the left below, there is a unique crisp filler. +A +X +B +Y +c +f +∀ +∀ +∃! +♭(XB) +♭(XA) +♭(Y B) +♭(Y A) +♭(◦ f) +♭(c◦) +♭(◦f) +♭(c◦) +⌟ +More formally, we write c ⊥♭ f for the proposition that the square on the right is a pullback. +Theorem 3.1.2. A crisp map f :: X → Y is ♯-modal if and only if for all crisp A, (ε : ♭A → A) ⊥♭ f. +Proof. If f is ♯-modal, then since ♯ is lex, it lifts on the right against all ♯-equivalences. For any crisp A, the ♭-counit +ε : ♭A → A is a ♯-equivalence by [47, Theorem 6.22]. Therefore, the square +XA +X♭A +Y A +Y ♭A +f◦ +◦ε +f◦ +◦ε +⌟ +is a pullback, and since ♭ preserves crisp pullbacks ([47, Theorem 6.10]), we see that ε ⊥♭ f. +On the other hand, suppose that f lifts crisply on the right against all ♭-counits. To show that f is ♯-modal, it +will suffice to show that its ♯-naturality square is a pullback. Let X → ♯X ×♯Y Y be the gap map of the ♯-naturality +square of f, seeking to show that this map is an equivalence. It suffices to split the gap map over the naturality +square, by the universal property of the pullback. So, consider the crisp square +♭(♯X ×♯Y Y) +X +♯X ×♯Y Y +Y +snd +ε +f +F +k +where F(t) :≡ (let t := p♭ in(fst p)♯) is a version of the first projection. To check that the square commutes, it +suffices by ♭-induction to give, for crisp elements u :: ♯X, y :: Y, and p :: (♯f(u) = y♯), a term of type f(u♯) = y. But +we have crisply that ♯f(u) ≡ ♯f(u♯♯) = (f(u♯))♯ by the definition of ♯f, and composing this path with p we know +12 + +p′ :: (f(u♯))♯ = y♯. By the lexness of ♯, we therefore also have p′′ :: ♯(f(u♯) = y), so that the square commutes by +p′′♯. +By hypothesis, there is a unique crisp map k : ♯X ×♯Y Y → X filling this square. The bottom triangle says +precisely that k lives over the second projection. We will turn the top triangle into a proof that k lives over the first +projection. +Let (u,y, p) : ♯X ×♯Y Y, seeking to show that k(u,y, p)♯ = u. This latter type of paths is codiscrete (because +♯X is codiscrete), and so when mapping into it we may assume by that u is of the form x♯, reducing our goal to +k(x♯,y, p)♯ = x♯. By the lexness of ♯, it suffices to give an element of ♯(k(x♯,y, p) = x), and for this it suffices to give +an element of k(x♯,y, p) = x under the hypotheses that x, y, and p are crisp. In this case, (x♯,y, p)♭ : ♭(♯X ×♯Y Y), +and so we have that k(x♯,y, p) = F((x♯,y, p)♭) by the upper triangle. But by definition, F((x♯,y, p)♭) ≡ x♯♯ ≡ x, so +that we have succeeded in giving the desired identification. +We have shown that k lives over the naturality square; now we need to show that it splits the gap map X → +♯X ×♯Y Y. To that end, consider the following diagram: +♭X +♭(♯X ×♯Y Y) +X +X +♯X ×♯Y Y +Y +gap +♭gap +ε +ε +f +f +k +Showing that the diagram commutes as drawn follows easily by ♭-induction. We then have two crisp fillers of the +outer square: first we have idX : X → X and k ◦gap : X → X. By the uniqueness of crisp fillers, we conclude that +these must be identical. +Knowing that the crisp ♯-modal maps may be characterized by lifting crisply against ♭-counits suggests that we +could axiomatize the particular qualities of ♯ by restricting the class of ♭-counits which it suffices to lift against. To +that end, we make the following definition. +Definition 3.1.3. Let G :: I → Type be a crisp type family indexed by a ♭-modal and inhabited type I. We say that +G detects continuity when, for every crisp map f :: X → Y, +{ f is ♯-modal} +�(ε : ♭Gi → Gi) ⊥♭ f +for all i :: I +� +Remark 3.1.4. Thinking externally, it is straightforward to see that any family Gi which generates a local topos +E in question under colimits will detect continuity for the focus given by the terminal map of toposes. This is +because ♭, as a left adjoint, commutes with all colimits; therefore the problem of lifting against the ♭-counit of any +object of E can be reduced to that of lifting against the ♭-counits of the generators Gi. +Lemma 3.1.5. A crisp type X is ♯-modal if and only if ♭(A → X) → ♭(♭A → X) is an equivalence for all crisp types +A, and if G detects continuity then it suffices to check for each Gi. +Proof. When f : X → 1, the square defining crisp lifting is a pullback iff the top map is an equivalence. +If a family detects continuity, then it is a separating family for crisp maps in the following precise sense. +Theorem 3.1.6. Suppose that G :: I → Type detects continuity. Let f :: X → Y be a crisp map for which ♭f : ♭X → +♭Y is an equivalence and for all i :: I, the induced map ♭(Gi → X) → ♭(Gi → Y) given by post-composing with f is +an equivalence. Then f is an equivalence. +13 + +Proof. First, note that f is a ♯-equivalence since it is by hypothesis a ♭-equivalence. It therefore suffices to show +that f is ♯-modal, which by the assumption that G detects continuity means showing that f lifts crisply against all +♭-counits ♭Gi → Gi for i :: I. Consider the following diagram: +♭(XGi) +♭(X♭Gi) +♭(♯XGi) +♭(Y Gi) +♭(Y ♭Gi) +♭(♯Y Gi) +♭(f◦) +♭(f◦) +∼ +∼ +♭(♯ f◦) +The square on the left is the one we need to show is a pullback. For this, it will suffice to show that the middle +map ♭(f◦) : ♭(X♭Gi) → ♭(Y ♭Gi) is an equivalence, since the leftmost vertical map is an equivalence by hypothesis +and any square with two sides equivalences is a pullback. +For that, the middle vertical map is equivalent by the adjunction ♭ ⊣ ♯ to the rightmost vertical map ([47, +Corollary 6.26]). But the rightmost vertical map is an equivalence because it is post-composition by the equivalence +♯f. +3.2 +Detecting Connectivity +A focus is said to be cohesive if ♭ has a further left adjoint S which is itself a modality: +♭(SA → X) ≃ ♭(A → ♭X). +This adjunction only determines S for crisp types. It is better to define S by nullifying a family of objects; then +S is determined for all types (of any size). To this end, we make the following definition. +Definition 3.2.1. Let G :: I → Type be a crisp type family indexed by a ♭-modal type. We say that G detects +connectivity when, for any crisp type X, +{X is ♭-modal} +�X is Gi-null +for all i :: I +� +If G detects connectivity, then S is defined to be nullification at the family Gi. +Remark 3.2.2. In Real Cohesion [47], the assertion that a given family G detects connectivity is known as Axiom +C0. In [44, Definition 5.2.48], a single object with this property is said to ‘exhibit the cohesion’. +If there is a family G which detects connectivity, then we say that the focus is cohesive. This is justified by the +following theorem, which we may import directly from Real Cohesion [47]. +Theorem 3.2.3. Suppose that G detects connectivity. Then a crisp type is S-modal if and only if it is ♭-modal, and +furthermore S is crisply left-adjoint to ♭: +♭(SA → X) → ♭(A → ♭X) +Proof. This is [47, Theorem 9.15]. +4 +Examples of Focuses +To keep the various operators visually distinct, we will use completely different symbols for each focus we are +interested in. The rules governing the type formers are unchanged. +• S ⊣ ♭ ⊣ ♯ denotes real cohesion, where a set of real numbers (possible the Dedekind reals or an axiomatically +asserted set of “smooth reals”) detects connectivity. +14 + +• re ⊣ sk0 ⊣ csk0 denotes simplicial cohesion, where the (axiomatically asserted) 1-simplex ∆[1] detects con- +nectivity. +• +< +⊣ +⊂ +⊣ +≺ +denotes global equivariant cohesion, where connectivity is detected by +≺ +BG for finite groups +G. This notational convention follows Sati and Schreiber [43]. +• Various topological toposes exhibit spatial type theory, with ♭ ⊣ ♯ retopologizing types with the discrete and +codiscrete topologies respectively. In particular, Johnstone’s topological topos has a focus whose continuity +is detected by the walking convergent sequence N∞, which may be constructed as the set of monotone +functions N → {0,1}. +4.1 +Real Cohesions +In Real Cohesion [47], Shulman gives the axiom R♭ which states that a crisp type is ♭-modal if and only if it is +RD-null, where RD is the set of Dedekind cuts. In the terminology of Definition 3.2.1, this says that RD detects +continuous real connectivity. +Axiom 1 (Continuous Real Cohesion). We assume that RD detects continuous real connectivity, and also Shul- +man’s Axiom T: For every x : RD, the proposition (x > 0) is ♯-modal. +Remark 4.1.1. Though Shulman does not consider this axiom, we may also add the assumption that the family +Rn +D detects continuous real continuity. Using this assumption, we may internalize the arguments of Example 8.33 +of [47] to show that the mysterious Axiom T follows from the proposition that if f :: Rn +D → RD is crisp and +f(x) > 0 for any crisp x :: Rn +D, then in fact f(x) > 0 for all (not necessarily crisp) x : Rn +D. Since, by Corollary +8.28 of [47] (assuming the crisp LEM or the axiom of countable choice), any crisp Dedekind real is a Cauchy real, +we are equivalently asking if a function f : RD → RD is positive on all Cauchy reals, is it always positive. This +seems obvious, but as Shulman notes in Example 8.34, this obvious statement is not always true; though assuming +countable choice it is likely provable. +There are continuous but non-differentiable functions f : RD → RD. If we want to work in a topos where the +types have a smooth structure instead of just a continuous structure, then we must work with a type of smooth reals +RS. The most common way to axiomatize the type of smooth reals is using the Kock-Lawvere axiom and the other +axioms of synthetic differential geometry. See, e.g. Section 4.1 of [36] for a list of these axioms. In any case, if +RS is a type of smooth reals, then we will take differential cohesion to mean that RS detects connectivity. +Axiom 2 (Differential Real Cohesion). If RS is a type of smooth reals (say, from synthetic differential geometry), +then we assume that RS detects differential connectivity. +4.2 +Simplicial Cohesion +There is a well known difficulty in describing simplicial types in ordinary homotopy type theory — the infinite +amount of coherence data is difficult to describe formally given the tools of type theory. This difficulty has led +to extensions of type theory such as two-level type theories [5, 8, 4] which augment HoTT with strict equalities +which can be use to define simplicial homotopy types satisfying the simplicial identities strictly, bypassing the +problematic tower of coherences. +Another approach to avoiding simplicial difficulties is to simply interpret type theory into a topos of simplicial +homotopy types, rather than mere homotopy types. This is the approach taken by Riehl and Shulman in [42], where +they present a type theory that makes every type into a simplicial type and has as primitives the simplices ∆[n], so +that a simplex in a type A is a function σ : ∆[n] → A. +In this section we will also work with simplicial homotopy types, but by different means to Riehl and Shulman’s +type theory. Instead, we will describe simplicial cohesion, with adjoint modalities re ⊣ sk0 ⊣ csk0. These are +defined semantically as follows: +15 + +• The (“simplicial flat”) 0-skeletal comodality X �→ sk0 X sends a simplicial type to its 0-skeleton: +... +X2 +X1 +X0 +sk0 +�−−−−−→ +... +X0 +X0 +X0 +• The (“simplicial sharp”) 0-coskeletal modality X �→ csk0 X sends a simplicial type to its 0-coskeleton: +... +X2 +X1 +X0 +csk0 +�−−−−−−→ +... +X0 ×X0 ×X0 +X0 ×X0 +X0 +• The (“simplicial shape”) realization modality X �→ reX sends a simplicial type to its realization (or homotopy +colimit), considered as a 0-skeletal simplicial type: +... +X2 +X1 +X0 +re· +�−−−−−→ +... +colimXn +colimXn +colimXn +Because simplicial sets are the classifying (0-)topos for strict intervals (totally ordered sets with distinct top +and bottom elements) [50], and since the ∞-topos of simplicial homotopy types is the enveloping ∞-topos3 of +simplicial sets [6], we may assume the existence of an interval ∆[1] to make sure that our type theory is interpreted +in an ∞-topos equipped with a geometric morphism to S ∆op. We may then define the n-simplex ∆[n] to be the +n-length increasing sequences in ∆[1], and define an n-simplex in a type X to be a map ∆[n] → X. +Axiom 3 (Simplicial Axioms). We presume that ∆[1] is a total order with distinct top and bottom elements which +we call 1 and 0 respectively. Explicitly, this means that we have elements 0, 1 : ∆[1] and a proposition x ≤ y : Prop +for x, y : ∆[1]. This order must satisfy the following axioms: +1. For all x, x ≤ x. +2. For all x, y, and z, if x ≤ y and y ≤ z then x ≤ z. +3. For all x and y, if x ≤ y and y ≤ x, then x = y. +4. For all x, y, either x ≤ y or y ≤ x. +5. For all x, 0 ≤ x and x ≤ 1. +6. 0 ̸= 1. +3The enveloping ∞-topos of a topos is its free (homotopy) cocompletion, fixing existing homotopy colimits. +16 + +From these axioms, we may define the other simplices ∆[n] to be the chains of length n in ∆[1]: +∆[n] :≡ {⃗x : ∆[1]n | x1 ≤ x2 ≤ ··· ≤ xn}. +We also assume the following: +• (Axiom ∆sk0) ∆[1] detects simplicial connectivity: a simplicially crisp type X is 0-skeletal if and only if +every map ∆[1] → X is constant. +• The family ∆[−] : N → Type detects simplicial continuity. +• (Axiom ∂∆) For i : ∆[1], we have csk0((i = 0)∨(i = 1)). +• Each ∆[n] is crisply projective. That is, for a simplicially crisp E : ∆[n] → Type, we have a map +sk0((i : ∆[n]) → ∃Ei) → ∃sk0((i : ∆[n]) → Ei). +As there is an obvious map the other way, this map is an equivalence. +Let us quickly set the stage by proving that the n-simplices have trivial geometric realization and the 0-skeleton +of the n-simplex ∆[n] is the ordinal [n] ≡ {0,...,n}. +Lemma 4.2.1. The order ∆[1] has finite meets and joins, and they distribute over each other. Moreover, the +inclusion {0,1} �→ ∆[1] is an inclusion of lattices. +Proof. Suppose that x,y : ∆[1]. Then either x ≤ y or y ≤ x. In the former case, define x∧y :≡ x and x∨y :≡ y, and +in the latter case x∧y :≡ y and x∨y :≡ x. If both hold, then x = y and the definitions agree. +If x,y,z : ∆[1], then these three may find themselves in any of 6 orderings. One may then check that in each of +these cases, meets distribute over joins and vice versa. For example, supposing that x ≤ y ≤ z, then x ∧ (y ∨ z) = +x∧z = x, while (x∧y)∨(x∧z) = x∨x = x. +Lemma 4.2.2. The n-simplex ∆[n] is a retract of the n-cube ∆[1]n. Moreover, the inclusion [n] �→ ∆[n] given by +i �→ 0 ≤ ··· ≤ 0 ≤ +i times +� +�� +� +1 ≤ ··· ≤ 1 +is a retract of the inclusion {0,1}n → ∆[1]n. +Proof. Given x1,...,xn : ∆[1], define m1 :≡ � +i:n xi and let i1 : n be its index, then m2 :≡ � +n\{m1} xi, and so on. Note +that m1 ≤ m2 ≤ ··· ≤ mn, so that m : ∆[n]. Finally, if the xi were already in increasing order, then mi = xi, showing +that this is a retract. +We note that this retract argument works just as well on {0,1}n → [n], if we identify [n] with the subset +{⃗x : {0,1}n | x1 ≤ ··· ≤ xn} of increasing sequences. Since it only makes use of the lattice structure of {0,1}n and +∆[1]n, and the inclusion is a lattice homomorphism, we conclude that the necessary squares commute. +Theorem 4.2.3. The n-simplex has trivial realization: re∆[n] = ∗. +Proof. The realization re∆[n] is a retract of the realization re∆[1]n, and this is contractible since ∆[1] detects +simplicial connectivity. +Theorem 4.2.4. The inclusion [n] �→ ∆[n] given by +i �→ 0 ≤ ··· ≤ 0 ≤ +i times +� +�� +� +1 ≤ ··· ≤ 1 +is a sk0-equivalence, showing that sk0 ∆[n] ≃ [n]. +17 + +Proof. We will show that [1] �→ ∆[1] is a sk0-equivalence. This will imply that [1]n �→ ∆[1]n is a sk0-equivalence; +since [n] �→ ∆[n] is a retract of this, we may conclude that it is a sk0-equivalence as well. +Since this inclusion {0,1} �→ ∆[1] is simplicially crisp, to show that it is a sk0-counit it will suffice to show that +it is a csk0-equivalence. We therefore need an inverse csk0 ∆[1] → csk0{0,1}. Since the codomain is 0-coskeletal, it +suffices to define this map on ∆[1]. So let i : ∆[1], seeking csk0{0,1}. By Axiom ∂∆, we have csk0((i = 0)∨(i = 1)), +and since our goal is 0-coskeletal, we may assume that i = 0 or i = 1. If i = 0, then we send it to 0csk0, if i = 1, +then we send it to 1csk0. +To show that this map is the inverse of csk0{0,1} → csk0 ∆[1], we may appeal to the fact that identities in a +modal type are modal, and so we may remove the csk0 around csk0((i = 0) ∨ (i = 1)) and check that the maps +invert each other on these elements, which they clearly do. +We can also define the type of n-simplices in a simplicially crisp type, and prove a few elementary lemmas +concerning the n-simplices of types. +Definition 4.2.5. Let X be a simplicially crisp type. Then define the type Xn of n-simplices in X as +Xn :≡ sk0(∆[n] → X). +If f : X → Y is a simplicially crisp map, then it induces a map fn : Xn → Yn by post-composition. +Lemma 4.2.6. Let f : X → Y be a simplicially crisp map. If fn : Xn → Yn is an equivalence for all n, then f is an +equivalence. +Proof. This is a special case of Theorem 3.1.6, noting that X0 ≃ sk0 X. +Lemma 4.2.7. Let f : X → Y be a simplicially crisp map. Then for a crisp y : ∆[n] → Y, we have +fib fn(ysk0) ≃ sk0((i : ∆[n]) → fib f (y(i))). +Proof. We compute: +fib fn(ysk0) ≡ (x : Xn)×(fnx = ysk0) +≡ (x : sk0(∆[n] → X))×let τsk0 := xin(f ◦τ)sk0 = ysk0 +≃ (x : sk0(∆[n] → X))×let τsk0 := xinsk0(f ◦τ = y) +≃ sk0((x : ∆[n] → X)×(f ◦x = y)) +≃ sk0((x : ∆[n] → X)×((i : ∆[n]) → (f(x(i)) = y(i)))) +≃ sk0((i : ∆[n]) → fib f (y(i))). +Lemma 4.2.8. Let f : X → Y be a simplicially crisp map. Then (im f)n ≃ im fn. +Proof. We use the projectivity of the simplices.4 +(im f)n ≡ sk0(∆[n] → (y : Y)×∃fib f (y)) +≃ (y : Yn)×let σsk0 := yinsk0((i : ∆[n]) → ∃fib f (σi)) +≃ (y : Yn)×let σsk0 := yin∃sk0((i : ∆[n]) → fib f (σi)) +≃ (y : Yn)×∃fib fn(y) +≡ im fn. +4This lemma is in fact equivalent to assuming the projectivity of the simplices. +18 + +The definition of the n-simplices that we gave above is simple, but it is not that straightforward to see that it is +functorial in the ordinal [n]. We can give an alternative definition of the n-simplices which makes the functoriality +evident. +Definition 4.2.9. Let Interval denote the category of intervals: totally ordered sets with distinct top and bottom. +The maps of Interval are the monotone functions preserving top and bottom. +Let FinOrd+ denote the category of finite inhabited ordinals and order preserving maps between them — the +usual “simplex category”. We denote by [n] the ordinal {0,...,n}. +We will need a standard reformulation of the category of finite ordinals in terms of intervals (see e.g. [32, +§VIII.7]) +Lemma 4.2.10. There is a contravariant, fully faithful functor ι : FinOrdop ++ → Interval sending [n] to [n+1] with +top element n+1 and bottom element 0. To a map f : [n] → [m], we define ι f : [m+1] → [n+1] by +ι f(i) :≡ +� +min{ j | i ≤ f(j)} +n+1 +if no such minimum exists. +Conversely, to a monotone map g : [m+1] → [n+1] preserving top and bottom, we define ι−1g : [n] → [m] by the +dual formula +(ι−1g)( j) :≡ max{i | g(j) ≤ i}. +We may now define the n-simplices in a way which makes clear their functoriality in the category of finite +inhabited ordinals. +Definition 4.2.11. We define the n-simplex ∆[n] to be +∆[n] :≡ Interval(ι[n],∆[1]). +Therefore, ∆ : FinOrd+ → Set gives a functor from finite inhabited ordinals to the category of sets, where ∆(f) : +∆[n] → ∆[m] is given by precomposing with ι f : [m+1] → [n+1]. +Noting that [n] ∼= Interval(ι[n],[1]) by the fully-faithfulness of ι, the inclusion of top and bottom elements [1] �→ +∆[1] induces a natural inclusion [n] �→ ∆[n] by post-composition. As we saw in Theorem 4.2.4, these inclusions are +sk0-counits. +4.2.1 +The ˇCech Complex +The 0-coskeleton modality csk0 is useful for working in simplicial cohesion since it enables us to give an easy +construction of the ˇCech complex of a map f : X → Y between 0-skeletal types. The ˇCech complex of such a map +is, externally speaking, the simplicial type formed by repeatedly pulling back f along itself: +ˇC(f) :≡ +... +X ×Y X ×Y X +X ×Y X +X +Definition 4.2.12. Let f : X → Y be a map. The ˇCech complex ˇC(f) of f is defined to be its csk0-image: +ˇC(f) :≡ (y : Y)×csk0((x : X)×(fx = y)). +19 + +We will justify this definition by calculating the type of n-simplices of ˇC(f) when both X and Y are 0-skeletal. +Proposition 4.2.13. Let f : X → Y be a simplicially crisp map between 0-skeletal types. Then +ˇC(f)n ≃ X ×Y ···×Y X ≃ (y : Y)×((x : X)×(fx = y))n+1 +is the (n+1)-fold pullback of f along itself. +Proof. We calculate: +ˇC(f)n :≡ sk0(∆[n] → ˇC(f)) +≡ sk0(∆[n] → (y : Y)×csk0((x : X)×(fx = y))) +≃ sk0((σ : ∆[n] → Y)×((i : ∆[n]) → csk0((x : X)×(fx = σi)))) +Since Y is 0-skeletal, any map ∆[n] → Y is constant, so we may continue: +≃ sk0((y : Y)×(∆[n] → csk0((x : X)×(fx = y)))) +Since, by Theorem 4.2.4, sk0 ∆[n] = [n], we may use the adjointness of sk0 and csk0 to continue: +≃ sk0((y : Y)×csk0([n] → (x : X)×(fx = y))) +Now, we may use Lemma 6.8 of [47] to pass the sk0 into the pair type, and then use that sk0 csk0 = sk0 to continue: +≃ ((u : sk0Y)×let u := ysk0 insk0([n] → (x : X)×(fx = y))) +However, all types involved are already 0-skeletal, so we may remove the sk0s: +≃ ((y : Y)×([n] → (x : X)×(fx = y))) +≃ (y : Y)×((x : X)×(fx = y))n+1 +This last type is the (n+1)-fold pullback of f along itself, displayed in terms of its diagonal map to Y. +We can prove modally that the realization of the ˇCech nerve of a map f : X → Y is the image im f of f. This +follows from Theorem 10.2 of Real Cohesion [47]. +Theorem 4.2.14. If A is 0-coskeletal, then reA is a proposition. As a corollary, re(csk0 X) ≃ ∥X∥. +Proof. In [47], this theorem is said to rely on the crisp Law of Excluded Middle. However a glance at the proof +reveals that this assumption is only used to assume the decidable equality of sk0 ∆[1]. Since we know that sk0 ∆[1] ≃ +{0,1} has decidable equality, the proof goes through without assuming crisp LEM. +Theorem 4.2.15. For a map f : X → Y, the realization re ˇC(f) of the ˇCech nerve is the realization reim f of the +image of f. If furthermore Y is 0-skeletal, then re ˇC(f) ≃ im f. +Proof. We compute: +re ˇC(f) ≡ re((y : Y)×csk0 fib f (y)) +≃ re((y : Y)×recsk0 fib f (y)) +≃ re((y : Y)× +��fib f (y) +��) +≡ reim f. +Now if Y is 0-skeletal, then by Lemma 8.17 of [47], im f is also 0-skeletal, since it is a subtype of a 0-skeletal type. +Therefore, reim f ≃ im f, so that in total re ˇC(f) ≃ im f. +20 + +As an application of ˇCech nerves, we can see how to extract coherence data for higher groups from their +deloopings. If we take the ˇCech nerve of the inclusion ptBG : ∗ → BG of the base point of the delooping of G, we +recover a simplicial type whose simplicial identities give coherences for the multiplication of G. +Proposition 4.2.16. Let G be a crisp, 0-skeletal higher group — a 0-skeletal type identified with the loops of a +pointed, 0-connected type BG. Then +ˇC(ptBG)n ≃ Gn. +Furthermore, d1 : ˇC(ptBG)2 → ˇC(ptBG)1 is the product of the projections d0 and d2 : G2 → G. +Proof. By Proposition 4.2.13, we know that +ˇC(ptBG)n ≃ (e : BG)×((x : ∗)×(ptBG∗ = e))n+1 +≃ (e : BG)×(ptBG = e)n+1 +≃ (e : BG)×(ptBG = e)×(ptBG = e)n +≃ (ptBG = ptBG)n += Gn. +In the second to last step, we contract (e : BG)×(ptBG = e). +Now, di : ˇC(ptBG)2 → ˇC(ptBG)1 is given by forgetting the ith component of the list (e,(a,b,c)) : (e : BG) × +(ptBG = e)n+1. Therefore, +d0(e,(a,b,c)) = (e,(b,c)) +d1(e,(a,b,c)) = (e,(a,c)) +d2(e,(a,b,c)) = (e,(a,b)) +Contracting away e and the first element of the pair, we get the three equations +d0(ba−1,ca−1) = cb−1 +d1(ba−1,ca−1) = ca−1 +d2(ba−1,ca−1) = ba−1 +and indeed, we have d1(ba−1,ca−1) = d0(ba−1,ca−1)d2(ba−1,ca−1). This is equivalent, but not quite the same, as +the standard presentation. It amounts to +d0(g,h) = hg−1 +d1(g,h) = h +d2(g,h) = g. +Using the ˇCech nerve, we can extract all the coherence conditions governing a homomorphism of higher +groups. We first note that the realization of the ˇCech nerve of a group is a delooping of it. +Proposition 4.2.17. Let G be a 0-skeletal higher group with simplicially crisp delooping BG, and let ˇC(G) be the +ˇCech nerve of the basepoint inclusion ptBG : ∗ → BG. Then the projection fst : ˇC(G) → BG is a re-unit. +Proof. By Theorem 5.9 of [34], BG is 0-skeletal. By Axiom ∆sk0, it is therefore re-modal. Therefore, to show +that fst : ˇC(G) → BG is a re-unit, it suffices to show that it is re-connected. Since BG is 0-connected, it suffices +to show that the fiber over the base point is re-connected, and this fiber is equivalent to csk0 G. This follows by +Theorem 4.2.14; recsk0 G is contractible. +21 + +Proposition 4.2.18. Let G and H be 0-skeletal higher groups. Then the type of homomorphisms G → H is +equivalent to the type of pointed maps ˇC(G) ·→ ˇC(H). +Proof. Recall that a homomorphism of higher groups is by definition a pointed map between their deloopings. +That is, a homomorphism ϕ : G → H is equivalently a diagram as on the left, while a pointed map between the +ˇCech nerves is a diagram as on the right: +� +� +� +� +� +� +� +� +� +� +� +∗ +∗ +BG +BH +ptBG +Bϕ +ptBH +� +� +� +� +� +� +� +� +� +� +� +?≃ +� +� +� +� +� +� +� +� +� +� +� +∗ +∗ +ˇC(G) +ˇC(H) +ptBG +f +ptBH +� +� +� +� +� +� +� +� +� +� +� +We are aiming for an equivalence between these two types, which we may present as a one-to-one correspondence. +So, to Bϕ : BG → BH and ptBG : ptBH = Bϕ(ptBG) and f : ˇC(G) → ˇC(H) and ptf : pt ˇC(H) = f(pt ˇC(G)) associate +the type +(□ : (x : ˇC(G)) → (Bϕ(fstx) = fst(fx)))×(ptBϕ ·□(pt ˇC(G)) = fst∗ ptf ) +which, diagrammatically, is the type of witnesses that the following diagram commutes: +∗ +∗ +ˇC(G) +ˇC(H) +BG +BH +f +Bϕ +fst +fst +To show that this gives a one-to-one correspondence means showing that the types of diagrams +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∗ +∗ +ˇC(G) +ˇC(H) +BG +BH +f +Bϕ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +and +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∗ +∗ +ˇC(G) +ˇC(H) +BG +BH +f +Bϕ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +are both contractible, the left for any homomorphism ϕ and the right for any pointed map f. +Let ϕ be a homomorphism. Since by definition ˇC(G) and ˇC(H) were the csk0-factorizations of the basepoint +inclusions, there is a unique filler of this square: +∗ +∗ +ˇC(H) +ˇC(G) +BG +BH +Bϕ +ptBG +ptBH +∃! +But this is precisely a rearrangement of the diagram on the left. +Similarly, if f is a pointed map, then re f : BG → BH makes the diagram on the right commute, and by the +universal property of the re-unit this is the unique such map. +22 + +4.3 +Global Equivariant Cohesion +In Global Homotopy Theory and Cohesion [41], Rezk shows that the ∞-topos of global equivariant homotopy types +is cohesive over the ∞-topos of homotopy types. While Rezk constructs his site out of all compact Lie groups, we +will follow Sati and Schreiber [43] in restricting our attention to the finite groups. The global orbit category Glo +is defined to be the full subcategory of homotopy types spanned by the deloopings BG of finite groups G. This +is a (2,1)-category, and the global equivariant topos is defined to be the ∞-category of homotopy type valued +presheaves on it. +There is an adjoint quadruple connecting the global equivariant topos and the topos of homotopy types: +S Gloop +S +colim +∆ +Γ +∇ +• colimX is the colimit of the functor X : Gloop → S , which takes the strict quotient of the global equivariant +homotopy type X. +• ∆S is the inclusion of constant functors: ∆X(BG) :≡ S. We will refer to such equivariant types as invariant +types. +• ΓX :≡ X(∗) is the evaluation at the point. This is known as the homotopy quotient of the global equivariant +homotopy type X. +• ∇S is the Yoneda embedding: ∇S(BG) :≡ S (BG,S). +This adjoint quadruple gives rise to the cohesive modalities +< +⊣ +⊂ +⊣ +≺ +of equivariant cohesion: +• The (“equivariant shape”) strict quotient modality X �→ +< +X sends a global equivariant type to its strict +quotient, considered as an invariant type. +• The (“equivariant flat”) homotopy quotient modality X �→ +⊂ +X sends a global equivariant type to its homotopy +quotient, considered as an invariant type. Internally speaking, we say that an equivariantly crisp type is +invariant when it is +⊂ +-modal. +• The (“equivariant sharp”) orbisingular modality X �→ +≺ +X sends a global equivariant type to its homotopy +quotient, but considered with its natural equivariance via maps from the deloopings of finite groups. +Our axioms for global equivariant cohesion are quite straightforward: +Axiom 4 (Global Equivariant Axioms). The type family +≺ +B : FinGrp → Type sending a finite group G to +≺ +BG +detects equivariant continuity and connectivity. +The types +≺ +BG for finite groups G are the orbi-singularities. By the definition above, we may recover X(BG) +(considered with its natural equivariance) as +≺ +( +≺ +BG → X). +Remark 4.3.1. The family +≺ +BG for finite groups G is a large family, but we may reduce it to a small family by +noting that the type of finite groups is essentially small. This is a useful observation, since it allows us to conclude +that +< +, defined by nullifying all +≺ +BG, is an accessible modality. +Remark 4.3.2. Global equivariant cohesion shares a feature with Shulman’s continuous real cohesion: both are +definable in the sense that the types which detect continuity and connectivity are definable without axioms in the +type theory. This is not the case for simplicial cohesion, which appears to require postulating the 1-simplex. It is +not clear to us whether there are any general features shared by definable cohesions. +23 + +In Proper Orbifold Cohomology [43], Sati and Schreiber work with equivariant differential cohesion to give +an abstract account of the differential cohomology of orbifolds. We can prove some of their lemmas easily in +global equivariant cohesion; we will return to prove the lemmas relating equivariant and differential cohesion in +the upcoming §6.2. +The following lemma appears as Proposition 3.62 in [43]. +Lemma 4.3.3. We have the following equivalences for the generic orbi-singularities +≺ +BG: +<≺ +BG ≃ ∗ +⊂≺ +BG ≃ BG +≺≺ +BG ≃ +≺ +BG. +Proof. The first equivalence follows by the assumption that +≺ +BG detects equivariant connectivity. The second +follows by combining Theorem 6.22 of [47] to see that +⊂< +BG ≃ +⊂ +BG with Theorem 5.9 of [34] to note that since +G is crisply +⊂ +-modal (as a finite set), BG is as well. The third is simply the idempotence of +≺ +. +The following lemma is a slight strengthening of Lemma 3.65 of [43]. +Lemma 4.3.4. Let X be an equivariantly crisp set. Then X is both invariant ( +⊂ +-modal) and orbi-singular ( +≺ +- +modal). +Proof. In both cases we will use that +≺ +BG detects equivariant continuity. To show that X is invariant, we must +show that the +⊂ +-counit is an equivalence. By Theorem 3.1.6, it suffices to show that the map +⊂ +( +≺ +BG → +⊂ +X) → +⊂ +( +≺ +BG → X) +is an equivalence. But +≺ +is a lex modality and BG is 0-connected; therefore, +≺ +BG is 0-connected. Furthermore, +⊂ +X is a set since X is, using Corollary 6.7 of [47]. Therefore, the above map is equivalent to +⊂ +ε : +⊂⊂ +X → +⊂ +X, +which is an equivalence. +Similarly, to show that X is orbi-singular, it suffices to show that the map +⊂ +( +≺ +BG → X) → +⊂ +(BG → X) +given by pre-composing with the +⊂ +-counit of +≺ +BG is an equivalence. But again, by the connectivity of +≺ +BG and +BG, this map is equivalent to the identity +⊂ +X → +⊂ +X, which is an equivalence. +Remark 4.3.5. Miller’s theorem (formerly the Sullivan conjecture) states that the space of maps BG → X with G +a finite group and X a finite cell complex is equivalent to X. In equivariant modal terms, this says that finite cell +complexes (the closure of the class {/0, ∗} under pushout) are +≺ +-modal. It is not likely that this theorem could be +proven on purely modal grounds. However, if Miller’s theorem were proven in ordinary HoTT, then the modal +statement could be proven in a manner similar to Lemma 4.3.4 but instead of appealing to the truncatedness of X, +appealing to the proof of Miller’s theorem (since any crisp finite cell complex is +⊂ +-modal as a crisp pushout of +⊂ +types). +4.4 +Topological Toposes +Johnstone defined his topological topos in [25] in order to provide a topos of spaces for which the geometric +realization of simplicial sets was a geometric morphism. The problem with using a real-cohesive topos for this +purpose (as suggested previously by Lawvere) is the failure of the analytic lesser limited principle of omniscience +which says that RD = (−∞,0]∪[0,∞) (see Theorem 11.7 of [47] and the discussion in Section 8.3 of ibid.). This +failure means that gluing together simplices along their (closed) faces gives the wrong topology on the resulting +space. +Johnstone remedies this by changing the test space from the real numbers to the walking convergent sequence +N∞. The walking convergent sequence may be defined internally as the set of monotone functions N → {0,1} (see +for example [20]). Therefore, it is rather straightforward to give an internal axiomatization for Johnstone’s topos. +24 + +Axiom 5 (Topological Focus). The topological focus is determined by asserting that N∞ detects topological conti- +nuity. +We may also be able to determine condensed homotopy types as in [19] — or rather the similar but more +topos-theoretic pyknotic homotopy types of [10] — using a similar axiom. Define a profinite set to be the limit of +a crisp diagram of finite sets indexed by a discrete (♭-modal) partially ordered set with decidable order. We may +then assert that the family of profinite sets detects condensed continuity. +However, we do not know if these axioms are sufficient for proving theorems in these topological toposes. +5 +Multiple Focuses +Now, we turn our attention to generalities on possible relationships between different focuses. For this section, fix +two focuses ♥ and ♣. First, we should show that the focuses do indeed commute: +Proposition 5.0.1. For any type B, the map ♯♥♯♣B → ♯♣♯♥B defined by +x �→ x♯♥♯♣ +♯♥♯♣ +is an equivalence. Furthermore, the maps ♯♥♯♣B → ♯♥♣B and ♯♣♯♥B → ♯♥♣B defined by +x �→ x♯♥♯♣ +♯♥♣ +x �→ x♯♣♯♥ +♯♥♣ +are also equivalences. +Proof. The first map is well-defined because the use of ♯♥- and ♯♣-introduction means that the assumption x +becomes crisp for both ♥ and ♣, so we may apply ♯♥- and then ♯♣-elimination to it. We may similarly define an +inverse by +x �→ x♯♣♯♥ +♯♣♯♥. +These maps are definitional inverses by the computation rules for ♯s. +The other maps are similarly well defined since being crisp for both ♥ and ♣ means being crisp for focus ♥♣. +The inverse may be defined in the straightforward way. +We also note that the ordering of focuses is reflected in the containment of their ♯-modal (and so also ♭-modal) +types. +Corollary 5.0.2. Suppose that ♣ ≤ ♥. Then any ♯♣-modal type is ♯♥-modal. +Proof. Since ♣ ≤ ♥ is defined to mean ♥♣ ≡ ♣, we know that ♯♥♣A ≡ ♯♣A. By assumption ♯♣A ≃ A, and +chaining this with the commutativity equivalence of Proposition 5.0.1 +♯♥A ≃ ♯♥♯♣A ≃ ♯♥♣A ≡ ♯♣A ≃ A. +Tracing these simple equivalences through, this does indeed give an inverse to the ♯♥-unit. +We can similarly show that the ♭s commute. This is made simpler through the use of crisp induction. +Proposition 5.0.3. Let A be an ♥♣-crisp type. Then ♭♥♭♣A → ♭♣♭♥A defined by +u �→ let v♭♥ := uin(let w♭♣ := vin(w♭♥)♭♣) +is an equivalence, natural in A. +25 + +Proof. In words, the map is defined as follows. Performing ♭♥-induction on u : ♭♥♭♣A gives an assumption v :♥ +♭♣A. A second induction on the term v : ♭♣A then gives us an assumption w :♥♣ A. This second induction is +‘♥-crisp ♭♣-induction’: the resulting assumption w inherits the ♥-crispness of term v and gains ♣-crispness from +the removal of ♭♣. Finally, we form (w♭♥)♭♣ by applying ♭-introduction twice. A map in the other direction is +constructed in the same way, and then the proofs these are inverse are immediate by another pair of inductions +each. +We note also that the ♭-inductions commute. +Lemma 5.0.4. ♭♥-induction and ♭♣-induction commute: +let u♭♣ := (let v♭♥ := M inN)inC = let v♭♥ := M in(let u♭♣ := N inC) +(when this is well-typed, i.e., v does not occur in C.) +Proof. First using uniqueness of ♭♣, we have +let u♭♣ := (let v♭♥ := M inN)inC += let w♭♥ := M in(let u♭♣ := (let v♭♥ := w♭♥ inN)inC) +≡ let w♭♥ := M in(let u♭♣ := N[w/v]inC) +≡ let v♭♥ := M in(let u♭♣ := N inC) +5.1 +Commuting Cohesions +Now let’s turn our attention to the relationships between two commuting cohesions. +Lemma 5.1.1. Suppose that ♥ and ♣ are both cohesive. If a ♥♣-crisp type A is ♭♥-modal, then S♣A is still +♭♥-modal. +Proof. We need to produce an inverse to the counit ε♥ : ♭♥S♣A → S♣A. First construct the composite +A s−→ ♭♥A +♭♥η♣ +−−−→ ♭♥S♣A +where A s−→ ♭♥A is the assumed inverse to ε♥ : ♭♥A → A. +The type ♭♥S♣A is certainly ♭♣-modal by commutativity of ♭♥ and ♭♣: +♭♣♭♥S♣A ≃ ♭♥♭♣S♣A ≃ ♭♥S♣A +and therefore the above map factors through i : S♣A → ♭♥S♣A, our purported inverse. +For one direction, consider the naturality square for the counit ε♥: +♭♥S♣A +♭♥♭♥S♣A +S♣A +♭♥S♣A +ε♥ +♭♥i +ε♥ +i +The map ♭♥i is equal to the comultiplication δ♥ : ♭♥A → ♭♥♭♥A defined by a♭♥ �→ a♭♥♭♥, because both are inverse +to the map ♭♥ε♥ : ♭♥♭♥A → ♭♥A. And so the bottom composite in the square is equal to ε♥ ◦ δ♥, which is the +identity. +26 + +In the other direction, because S♣A is S♣-modal, it suffices to show that the composite +A → S♣A → ♭♥S♣A → S♣A +is equal to the unit η♣ : A → S♣A. For this we have the following commutative diagram: +A +♭♥A +A +S♣A +♭♥S♣A +S♣A +∼ +η♣ +♭♥η♣ +ε♥ +η♣ +i +ε♥ +The left square commutes by the definition of i, the right square by naturality of ε♥, and the composite along the +top is the identity. +Lemma 5.1.2. Suppose that ♥ and ♣ are both cohesive. Then S♥S♣A → S♣S♥A is an equivalence for any ♥♣-crisp +type A. +Proof. The map η♣ ◦η♥ : A → S♥A → S♣S♥A factors through S♥S♣A, because S♣S♥A is ♭♣-modal, as a type of the +form ♭♣X, and also ♭♥-modal, by the previous lemma. The map the other way is defined similarly. To show these +are inverses it suffices to show that they become so after precomposition with the composites of the units, because +S♣S♥A and S♥S♣A are ♥♣-discrete; this is immediate by the definition of the maps. +We might also hope that, say, ♭♥ and S♣ commute in general, but there is a useful sanity check that shows this +is not possible. In the bare type theory with no axioms, there is nothing that prevents interpretation in a model +where ♭♥ ≡ ♭♣ and S♥ ≡ S♣. In ordinary cohesive type theory it is certainly not the case that ♭ and S commute, and +so ♭♥S♣ ≃ S♣♭♥ cannot be provable without further assumptions on ♥ and ♣. +A sufficient assumption on our focuses to make ♭♥ and S♣ commute in this way is the following: +Definition 5.1.3. Suppose that G :♥♣ I → Type and H :♥♣ J → Type detect ♥ and ♣ connectivity respectively. +We say that focuses ♥ and ♣ are orthogonal if Gi is ♭♣-modal for all i, and Hj is ♭♥-modal for all j. +Our present goal is to show that this indeed makes ♭♥S♣ ≃ S♣♭♥. We will in fact only use that the Gi are +♭♣-modal; of course the dual results, flipping ♥ and ♣, require the other half of orthogonality. +Lemma 5.1.4. Let ♥ and ♣ be cohesive focuses that are orthogonal. Then for any ♣-crisp A, if A is S♥-modal, +♭♣A is still S♥-modal. +Proof. Our goal is to show that ♭♣A is equivalent to Gi → ♭♣A via precomposition by Gi → 1, for any i : I. We +easily check that the type Gi → ♭♣A is ♣-discrete: for any Hj, +Hj → (Gi → ♭♣A) ≃ Hj → (Gi → ♭♣A) +≃ Gi → (Hj → ♭♣A) +≃ Gi → ♭♣A +because ♭♣A is ♣-discrete. Then, by adjointness of S♣ and ♭♣: +(Gi → ♭♣A) ≃ ♭♣(Gi → ♭♣A) +≃ ♭♣(S♣Gi → A) +≃ ♭♣(Gi → A) +≃ ♭♣A +27 + +Proposition 5.1.5 (Crisp S♥-induction). Suppose that ♥ and ♣ are cohesive and orthogonal. If B is ♥-discrete and +♣-crisp, then for any ♣-crisp A the map +♭♣(♭♣S♥A → B) → ♭♣(♭♣A → B) +given by precomposition by ♭♣η♥ : ♭♣A → ♭♣S♥A is an equivalence. +Remark 5.1.6. As a rule, crisp S-induction would be written: +♣\Γ,x :♣ S♥A ⊢ C +♣\Γ,x :♣ S♥A ⊢ w : is-♭♥-modal(C) +♣\Γ ⊢ M : S♥A +♣\Γ,u :♣ A ⊢ N : C[uS♥/x] +Γ ⊢ (let uS♥ := M inN) : C[M/x] +Proof. We deploy the usual trick for deriving crisp induction principles: using ♯♣ to move the ♭♣ out of the way. +Crucially, the previous proposition is what allows us to apply the universal property of S♥ on maps into ♯♣B. +♭♣(♭♣S♥A → B) ≃ ♭♣(S♥A → ♯♣B) +≃ ♭♣(A → ♯♣B) +≃ ♭♣(♭♣A → B) +Proposition 5.1.7. If ♥ and ♣ are orthogonal and cohesive focuses, then S♥ and ♭♣ commute on ♣-crisp types. +Proof. This is now straightforward induction on S♥ and ♭♣ in both directions, using crisp S♥-induction when +defining ♭♣S♥A → S♥♭♣A and Lemma 5.1.4 when defining S♥♭♣A → ♭♣S♥A to know that ♭♣S♥A is ♥-discrete. +Corollary 5.1.8. If ♥ and ♣ are orthogonal and cohesive focuses, then ♭♥ and ♯♣ commute on ♥♣-crisp types. +Proof. On ♥♣-crisp types, ♭♥♯♣ is right adjoint to ♭♣S♥, and ♯♣♭♥ is right adjoint to S♥♭♣. Therefore, if ♭♣S♥X ≃ +S♥♭♣ for a ♥♣-crisp type X, then also ♭♥♯♣X ≃ ♯♣♭♥. +Next we investigate a relationship between S and ♯. Again, this depends on a relationship between the families +which detect continuity and connectivity of the two focuses. +Proposition 5.1.9. Suppose that ♣ is cohesive, and that the following hold: +1. H :♥♣ I → Type detects ♣-connectivity, and I is ♭♥-modal. +2. Hi is ♭♥-modal for all i :♥ I. (In particular, if ♥ and ♣ are orthogonal) +Then if X is S♣-modal then ♯♥X is also S♣-modal. +Proof. Since H detects ♣-connectivity, it suffices to show that ♯♥S♣X is Hi-null for every i : I. Since I is ♭♥-modal, +we may assume that i :♥ I is ♥-crisp by ♭♥-induction. Then we can compute: +(Hi → ♯♥X) ≃ ♯♥(Hi → ♯♥X) +≃ ♯♥(♭♥Hi → X) +≃ ♯♥(Hi → X) +since Hi was assumed ♭♥-modal +≃ ♯♥X +since X is S♣-modal +Tracing upwards through this series of equivalences shows that the composite is indeed the inclusion of constant +functions. +28 + +On the opposite extreme of orthogonality, we can see that if the Gi which detect the connectivity of ♥ are +S♣-connected, then any S♣-modal type is S♥-modal. +Proposition 5.1.10. Suppose that ♥ and ♣ are cohesive focuses where G : I → Type detects the connectivity of +♥. Then the following are equivalent: +1. Every Gi is S♣-connected. +2. Any S♣-modal type is S♥-modal. +Proof. Suppose that Gi is S♣-connected for all i and that X is S♣-modal. We may compute: +(1 → X) ≃ (S♣Gi → X) ≃ (Gi → X) +In the first equivalence, we use that Gi is S♣-connected, and in the second that X is S♣-modal. We conclude that X +is S♥-modal. +Conversely, suppose that any S♣-modal type is S♥-modal. Then in particular S♣Gi is S♥-modal, so that the +identity map S♣Gi → S♣Gi factors through S♥S♣Gi. But by Lemma 5.1.2 we have +S♥S♣Gi ≃ S♣S♥Gi ≃ S♣∗ ≃ ∗. +Therefore, the identity of S♣Gi factors through the point, which means it is contractible. +6 +Examples with Multiple Focuses +In this section, we will see examples with multiple focuses. In particular, we will see simplicial real cohesion, +equivariant differential cohesion, and supergeometric cohesion. +6.1 +Simplicial Real Cohesion +We assume two basic focuses: the real (continuous or differential) focus S ⊣ ♭ ⊣ ♯, and the simplicial focus re ⊣ +sk0 ⊣ csk0. We will write R for whichever flavor of real numbers is used in the real cohesive focus. +We will assume both the axioms of real cohesion and simplicial cohesion, as well as the following axiom +relating the two focuses. +Axiom 6 (Simplicial Real Cohesion). We assume that the real focus and simplicial focus are orthogonal — which +is to say, R is 0-skeletal and that ∆[1] is discrete. Furthermore, we assume that S is computed pointwise: for any +simplicially crisp type X, the action (ηS)n : Xn → (SX)n of the S-unit of X on n-simplices is itself a S-unit. +Our goal in this section will be to prove that if M is a 0-skeletal type — to be thought of as a “manifold”, having +only real-cohesive structure but no simplicial structure — and U is a good cover of M — one for which the finite +intersections are S-connected whenever they are inhabited — then the homotopy type SM of M may be constructed +as the realization of a discrete simplicial set — namely, the ˇCech nerve of the open cover, with each open replaced +by the point. +Definition 6.1.1. Let M be a 0-skeletal type. A cover of M consists of a discrete 0-skeletal index set I, and a +family U : I → (M → Prop) of subobjects of M so that for every m : M there is merely an i : I with m ∈ Ui. We +may assemble a cover into a single surjective map c : � +i:IUi → M, where +� +i:I +Ui :≡ (i : I)×(m : M)×(m ∈ Ui). +A cover U : I → (M → Prop) is a good cover if for any n : N and any k : [n] → I, the S-shape of the intersection +� +i:[n] +Uk(i) :≡ (m : M)×((i : [n]) → (m ∈ Uk(i))). +is a proposition. That is, S(Uk(0) ∩···∩Uk(n)) is contractible whenever there is an element in the intersection. +29 + +We begin with a few ground-setting lemmas. +Lemma 6.1.2. Let U : I → (M → Prop) be a simplicially crisp cover, and let c : � +i:IUi → M be the associated +covering map. Consider the projection π : ˇC(c) → csk0 I defined by (m,z) �→ (fstzcsk0)csk0. Over a simplicially +crisp n-simplex k : ∆[n] → csk0 I, we have +fibπn(ksk0) ≃ +� +i:[n] +Uk(isk0). +As a corollary, we have that +ˇC(c)n ≃ (k : I[n])× +� +i:[n] +Uk(i). +Proof. We compute: +fibπn(ksk0) ≡ (x : ˇC(c)n)×(πnx = ksk0) +≃ ((m,z) : (m : M)×((i : I)×(m ∈ Ui))[n])×(πne(m,z) = ksk0) +≃ (m : M)×(K : [n] → I)×(p : (i : [n]) → (m ∈ UK(i)))×(πne(m,i �→ (K(i), p)) = ksk0) +Here, e(m,z) is image under the equivalence from Proposition 4.2.13. When all the modal dust settles, we will be +left knowing that πne(m,i �→ (K(i), p)) : sk0(∆[n] → csk0 I) is the unique correspondent to K : [n] → I under the +sk0 ⊣ csk0 adjunction. Therefore, we may contract K away with ksk0 in the above type to get: +≃ (m : M)×((i : [n]) → m ∈ Uk(isk0)). +For the next lemma, we will need to know that S commutes with csk0 on suitably crisp types. +Theorem 6.1.3. For any simplicially crisp type X, we have that Scsk0 X ≃ csk0 SX. +Proof. We know by Proposition 5.1.9 that csk0 SX is S-modal. We therefore have a map Scsk0 X → csk0 SX given +as the unique factor of csk0(−)S : csk0 X → csk0 SX. We will show that this map is an equivalence. Since it is crisp, +it suffices to show that it is an equivalence on n-simplices. To that end, we compute: +sk0(∆[n] → Scsk0 X) ≃ Ssk0(∆[n] → csk0 X) +≃ Ssk0([n] → X) +≃ sk0(S([n] → X)) +≃ sk0([n] → SX) +≃ sk0(∆[n] → csk0 SX) +It remains to show that this is indeed the right equivalence. Since the first equivalence in the series above is given +as the inverse of (−)S +n : (csk0 X)n → (Scsk0 X)n, it suffices to check that given a crisp z : ∆[n] → csk0 X, (zsk0)S +corresponds under the above equivalences to (csk0(−)S ◦z)sk0. First, we send (zsk0)S to ((z◦(−)sk0)sk0)S. Then, we +send it to ((z◦(−)sk0)S)sk0, and then to (i �→ (z(isk0)S))sk0. Finally, we map this to (i �→ ((z(isk0sk0)S))csk0 sk0, which +does equal (csk0(−)S ◦z)(i) ≡ (z(i)S)csk0 at i : ∆[n]. +Lemma 6.1.4. Let U : I → (M → Prop) be a simplicially crisp cover, and let c : � +i:IUi → M be the covering map +itself. Consider the “projection” π : ˇC(c) → csk0 I defined by (m,z) �→ (fstzcsk0)csk0. Then U is a good cover if and +only if the restriction π : ˇC(c) → imπ is a S-unit. +30 + +Proof. Since csk0 and S commute by Theorem 6.1.3 and I is discrete, csk0 I is also discrete. As the subtype +of a discrete type, imπ is discrete. Therefore, it suffices to show that π : ˇC(c) → imπ induces an equivalence +S ˇC(c) ∼−→ imπ if and only if the cover U is good. Since π is crisp, S ˇC(c) → imπ is an equivalence if and only if it is +an equivalence on all n-simplices. On n-simplices, this map (at the top of the following diagram) is equivalent to +the map on the bottom of the following diagram: +(S ˇC(c))n +(imπ)n +S(ˇC(c)n) +imπn +S +� +(k : I[n])× � +i:[n]Uk(i) +� +(k : I[n])×S +�� +i:[n]Uk(i) +� +(k : I[n])×∃� +i:[n]Uk(i) +The bottom map is an equivalence if and only if the cover is good, and so we conclude the same for the top +map. +Finally, we can piece these lemmas together for our result. +Theorem 6.1.5. Let U : I → (M → Prop) be a simplicially crisp good cover of a 0-skeletal type M. Let π : ˇC(U) → +csk0 I be the projection. Then +reimπ ≃ SM. +This exhibits the shape SM as the realization of a discrete (simplicial) set. +Proof. Since the cover is good, we have that imπ ≃ SˇC(U), so that +reimπ ≃ reSˇC(U) ≃ Sre ˇC(U) ≃ SM. +Now, since I is a set and csk0 is a lex modality, csk0 I is also a set and so imπ is a set as well. Furthermore, since +subtypes of discrete types are discrete by Lemma 8.17 of [47], and csk0 I is discrete since by Theorem 6.1.3 S and +csk0 commute on simplicially crisp types, imπ is discrete. +6.2 +Equivariant Differential Cohesion +In Proper Orbifold Cohomology, Sati and Schreiber work in equivariant differential cohesion to describe the differ- +ential cohomology of orbifolds. This cohesion involves both the equivariant focus +< +⊣ +⊂ +⊣ +≺ +and the differential +(real-cohesive) focus S ⊣ ♭ ⊣ ♯. In this section, we will assume both the axioms of equivariant cohesion and differ- +ential real cohesion. We will refer to the smooth reals by R. +Unlike the simplicial real cohesive case, we do not need to add additional axioms to ensure that the equivariant +and differential cohesion are orthogonal. +Lemma 6.2.1. Equivariant and differential cohesion are orthogonal. That is: +1. The smooth reals R are invariant ( +⊂ +-modal). +2. For any finite group G, +≺ +BG is discrete (♭-modal). +Proof. Since the smooth reals are a set, they are invariant by Lemma 4.3.4. Similarly, since BG is discrete and +hence S-modal (by Theorem 5.9 of [34]), +≺ +BG is still S-modal by Proposition 5.1.9. +31 + +The following lemma appears as Lemma 3.67 of [43], and is proven quickly with our general lemmas concern- +ing orthogonal cohesions. +Lemma 6.2.2. Suppose that X is both differentially and equivariantly crisp. Then +⊂ +SX ≃ S +⊂ +X +⊂ +♭X ≃ ♭ +⊂ +♭X +⊂ +♯X ≃ ♯ +⊂ +X. +Proof. The first equivalence follows by Proposition 5.1.7. The second equivalence follows by Proposition 5.0.3. +The third follows by Corollary 5.1.8. +6.3 +Supergeometric Cohesion +In his habilitation, Differential Cohomology in a Cohesive ∞-Topos [44], Schreiber describes an increasing tower +of adjoint modalities which appear in the setting of supergeometry. The setting for supergeometric cohesion — +called “solid cohesion” in ibid. — is sheaves on the opposite of a category of super C ∞-algebras. Schreiber calls +these sheaves super formal smooth ∞-groupoids. Specifically, the site is (the opposite of) the full subcategory of +the category of super commutative real algebras spanned by objects of the form +C ∞(Rn)⊗W ⊗ΛRq +where W is a Weil algebra — a commutative nilpotent extension of R which is finitely generated as an R-module. +The factor C ∞(Rn)⊗W is even graded, while the Grassmannian ΛRq is odd graded. See Definition 6.6.13 of [44]. +The inclusion of algebras of the form C ∞(Rn) ⊗W has a left and a right adjoint. The left adjoint is given +by projecting out the even subalgebra, and the right adjoint is given by quotienting by the ideal generated by the +odd graded elements. This gives rise to an adjoint quadruple between the resulting toposes of sheaves and thus an +adjoint triple of idempotent adjoint (co)monads on the topos of super formal smooth ∞-groupoids: ⇒ ⊣ ⇝ ⊣ Rh. +Of these, ⇒ and Rh are idempotent monads. However, ⇒ does not preserve products, and so does not give an +internal modality. The action of Rh is easy to define: +RhX(C ∞(Rn)⊗W ⊗ΛRq) := X(C ∞(Rn)⊗W). +That is, RhX is defined by evaluating at the even part of the superalgebra in the site. We may characterize it +internally by localizing at the odd line R0|1, which is the sheaf represented by the free superalgebra on one odd +generator ΛR. We turn to the internal story now. +The topos of super formal smooth ∞-groupoids also supports the differential real cohesive modalities S ⊢ ♭ ⊢ ♯. +These destroy all geometric structure — super and otherwise. For this reason, we will work with the lattice +{diff < super < ⊤} of focuses. +The modalities of the super focus are ⇝ ⊢ Rh. We will refer to +⇝ +X as the even part of X, while Rh is known as +the rheonomic modality. We assume the following axioms for supergeometric or solid cohesion. +Axiom 7 (Solid Cohesion). Solid cohesion uses the focus lattice {diff < super < ⊤}. We use the definition of real +superalgebras due to Carchedi and Roytenberg [15]. +1. We assume a commutative ring R1|0 satisfying the axioms of synthetic differential geometry (as e.g. in +Section 4.1 of [36]) known as the smooth reals or the even line. +2. We assume an R1|0-module R0|1. There is furthermore a bilinear multiplication R0|1×R0|1 → R1|0 which sat- +isfies a2 = 0 for all a : R0|1. Together these axioms imply that R1|1 := R1|0×R0|1 is a R1|0-supercommutative +superalgebra. +3. We assume the following odd form of the Kock-Lawvere axiom: For any function f : R0|1 → R0|1 with +f(0) = 0, there is a unique r : R1|0 with f(x) = rx for all x. +4. We assume that R0|1 is ⇝-connected. +32 + +5. We assume that R1|0 detects differential connectivity. +6. We assume that a type is Rh-modal if and only if it is R0|1-null. +Remark 6.3.1. It might seem prudent to instead ask that differential connectivity is detected by the family con- +sisting of both R1|0 and R0|1, since we want S to nullify all representables Rn|q, but it suffices to test with R1|0 since +R0|1 admits an explicit contraction by its R1|0-module structure (appealing to Lemma 6.10 of [34]). +By Corollary 5.0.2, any ♯-modal type is Rh-modal. But also every ♭-modal type is Rh-modal. +Lemma 6.3.2. If X is S-modal (and, in particular, if X is ♭-modal), then X is Rh-modal. +Proof. 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URL: http://www.numdam.org/item/CTGDC_1993__34_4_259_0/. +35 + +A +Proof Sketches for Admissible Rules +We sketch proofs that the operations on syntax are admissible, demonstrating the interesting cases; those that +involve division (CTX-EXT, ♭-FORM/INTRO/ELIM, ♯-ELIM) or promotion (♯-FORM/INTRO). +Definition A.0.1. The ♥Γ and ♥\Γ context operations extend in the obvious way to telescopes Γ′, so that +♥(Γ,Γ′) ≡ (♥Γ),(♥Γ′) +♥\(Γ,Γ′) ≡ (♥\Γ),(♥\Γ′) +Lemma A.0.2 (Weakening). Single-variable weakening is admissible. +WK +Γ,Γ′ ⊢J +Γ,w :♣ W,Γ′ ⊢J +−−−−−−−−−−− +Moreover, weakening does not change the size of the derivation tree. +Proof. Induction onJ . +Case (division). +♭-FORM ♥\(Γ,w :♣ W,Γ′) ⊢ A type +Γ,w :♣ W,Γ′ ⊢ ♭♥A type +There are two subcases: +• If ♣ ≤ ♥: then ♥\(Γ,w :♣ W,Γ′) ≡ (♥\Γ),w :♣ W,(♥\Γ′), in which case we induct on the type A +and reapply the rule. +• If ♣ ̸≤ ♥: then ♥\(Γ,w :♣ W,Γ′) ≡ (♥\Γ),(♥\Γ′) ≡ ♥\(Γ,Γ′), and so already ♥\(Γ,w :♣ W,Γ′) ⊢ +A type and we can reapply the rule. +Case (promotion). +♯-FORM ♥(Γ,w :♣ W,Γ′) ⊢ A type +Γ,w :♣ W,Γ′ ⊢ ♯♥A type +By definition ♥(Γ,w :♣ W,Γ′) ≡ ♥Γ,w :♥♣ W,♥Γ′, and so we induct on A (now weakening with a variable +of focus ♥♣). +Lemma A.0.3. For any two focuses ♥ and ♣, the context ♣\(♥Γ) is an iterated weakening of ♥(♣\Γ). +Proof. This can be checked variablewise. Given a variable x :♠ A in Γ, if it survives to ♥(♣\Γ) as x :♥♠ A, then we +must have ♠ ≤ ♣. Multiplying by ♥, it follows that ♥♠ ≤ ♥♣ ≤ ♣, and so x :♥♠ A also occurs in ♣\(♥Γ). +Lemma A.0.4. The following equations involving contexts and telescopes hold: +(♥Γ′)[s/z] ≡ ♥(Γ′[s/z]) +(♥\Γ′)[s/z] ≡ ♥\(Γ′[s/z]) +36 + +Lemma A.0.5 (Substitution). +SUBST +♣\Γ ⊢ s : S +Γ,z :♣ S,Γ′ ⊢J +Γ,Γ′[s/z] ⊢J [s/z] +−−−−−−−−−−−−−−−−−−−− +Proof. Induction onJ . +Case (variable). +Three subcases as usual, for x ∈ Γ, x ≡ z and x ∈ Γ′. The interesting one is x ≡ z: +VAR Γ,z :♣ S,Γ′ ⊢ s : S +Applying DIVIDE-WK to ♣\Γ ⊢ s : S gives Γ ⊢ s : S, which can be further weakened to Γ,Γ′ ⊢ s : S. +Case (division). +♭-FORM ♥\(Γ,z :♣ S,Γ′) ⊢ A type +Γ,s :♣ S,Γ′ ⊢ ♭♥A type +There are two subcases: +• If ♣ ≤ ♥: then ♥ \ (Γ,z :♣ S,Γ′) ≡ (♥ \ Γ),z :♣ S,(♥ \ Γ′), in which case we induct, getting (♥ \ +Γ),(♥\Γ′)[s/z] ⊢ A[s/z] type. This context is equal to ♥\(Γ,Γ′[s/z]) ctx, so we can reapply the rule. +• If ♣ ̸≤ ♥: then ♥ \ (Γ,z :♣ S,Γ′) ≡ (♥ \ Γ),(♥ \ Γ′) ≡ ♥ \ (Γ,Γ′), and so z does not occur in A, and +A ≡ A[s/z], in which case we may reapply the rule. +Case (promotion). +♯-FORM ♥(Γ,z :♣ S,Γ′) ⊢ A type +Γ,z :♣ S,Γ′ ⊢ ♯♥A type +By definition ♥(Γ,s :♣ S,Γ′) ≡ ♥Γ,s :♥♣ S,♥Γ′. Applying PROMOTE to ♣\Γ ⊢ s : S yields ♥(♣\Γ) ⊢ s : S. +By Lemma A.0.3, this can be weakened to ♣(♥\Γ) ⊢ s : S, whose context is equal to (♥♣)\(♥Γ), which +is of the correct shape to be substituted into ♥Γ,z :♥♣ S,♥Γ′ ⊢ A type. Substitution gives ♥Γ,(♥Γ′)[s/z] ⊢ +A[s/z] type, and this context is equal to ♥(Γ,Γ′[s/z]) ctx, so we can reapply the rule. +Lemma A.0.6 (Promote). +PROMOTE-CTX +Γ ctx +♣Γ ctx +−−−− +PROMOTE +Γ ⊢J +♣Γ ⊢J +−−−−− +Proof. First, PROMOTE-CTX is by induction on the length of the context. Consider a context Γ,x :♥ A ctx, so that +♥ \ Γ ⊢ A type. Applying PROMOTE to A gives ♣(♥ \ Γ) ⊢ A type, which can be weakened to (♣♥) \ (♣Γ) ⊢ +A type by Lemma A.0.3, letting us form the context ♣Γ,x :♣♥ A ctx. +Case (variable). +The variable rule is immediate, because modifying the annotation on a variable does not change whether it +is usable. +37 + +Case (division). +♭-FORM ♥\Γ ⊢ A type +Γ ⊢ ♭♥A type +Inductively ♣(♥\Γ) ⊢ A type, which can be weakened to ♥\(♣Γ) ⊢ A type, and we reapply the rule. +Case (promotion). +♯-FORM ♥Γ ⊢ A type +Γ ⊢ ♯♥A type +Inductively ♣(♥Γ) ⊢ A type, and ♣(♥Γ) ≡ ♥(♣Γ), so we may reapply the rule. +Lemma A.0.7 (Division). +DIVIDE-CTX +Γ ctx +♣\Γ ctx +−−−−−− +DIVIDE-WK +Γ ctx +♣\Γ ⊢J +Γ ⊢J +−−−−−−−−−−−− +Proof. First DIVIDE-CTX. Consider a context Γ,x :♥ A ctx, so that ♥\Γ ⊢ A type. There are two cases: +• If ♥ ≤ ♣: Then +♥\Γ ≡ (♥♣)\Γ ≡ ♥\(♣\Γ), +and so ♥\(♣\Γ) ⊢ A type is of the right shape to form ♣\Γ,x :♥ A ctx. +• If ♥ ̸≤ ♣: Then ♣\(Γ,x :♥ A) ≡ ♣\Γ is well-formed by induction. +Now DIVIDE-WK. On terms: +Case (variable). +If x :♥ A is in context ♣\Γ, then it must also be in Γ and so we may reuse the variable rule. +Case (division). +♭-FORM ♥\(♣\Γ) ⊢ A type +♣\Γ ⊢ ♭♥A type +We know ♥\(♣\Γ) ≡ ♣\(♥\Γ), and so inductively ♥\Γ ⊢ A type, and we can reapply the rule. +Case (promotion). +♯-FORM ♥(♣\Γ) ⊢ A type +♣\Γ ⊢ ♯♥A type +♥(♣ \ Γ) ⊢ A type may be weakened to ♣ \ (♥Γ) ⊢ A type (without increasing the size of the derivation). +Inductively ♥Γ ⊢ A type, and we can reapply the rule. +38 +