Commuting Cohesions

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 Čech 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.

Semantics of multimodal adjoint type theory

“…(The left adjoint in this case also has a further pair of left adjoints.) Example 6.13 If L is a meet-semilattice, regarded as a monoidal poset and thereby a one-object 2category, we obtain an instance of MATT that is similar to the left-lifting theory of [31]. In many of their examples each "focus" is cohesive, hence the further left adjoints needed for locks already exist.…”

“…(ii) For any 2-category L, let M = L[ † L] be obtained by formally adjoining a left adjoint † µ to each µ in L. We take only identities to be tangible, sharp, and transparent, and the sinister morphisms to be these left adjoints † µ; then MATT reduces to FitchTT [11] over L with actual left adjoints. (iii) The closest match with theories such as [27,37,31] occurs when M = L[L † ] is obtained by formally adjoining a right adjoint µ † to each morphism µ of L. In this case we take the tangible, sharp, and sinister morphisms to be the image of L in L[L † ]; thus all the modal operators come in adjoint pairs. Different theories make different choices about transparency: in [37] only identities are transparent, while in [31] the transparent morphisms are also the image of L. But in fact, if a morphism is both sinister and tangible, then it "might as well" be transparent, in that elimination rules with it as framing can be deduced from those with identity framing; the proof follows [37,Lemma 5.1].…”

“…Many local toposes are also cohesive, meaning that the leftmost adjoint of their adjoint triple has a further left adjoint. This left adjoint is not usually finitely continuous, so it cannot be represented internally as a judgmental modality with co-dextrification, although it can be introduced axiomatically as in [37,31]. When it exists, co-dextrification is not needed to interpret the other modalities.…”

“…Instead, the spiritual generalizations of split-context theories, sometimes called left-division theories (e.g. [32,31]), annotate each context variable with a morphism of M that is implicitly applied to it, and the modal rules modify these annotations. In our simple example, each variable in a q-context is annotated with µ or 1 q , and the operation Γ/µ deletes the 1 q -annotated variables and uses the others to form a p-context.…”