Brouwer's fixed-point theorem in real-cohesive homotopy type theory

Abstract

We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of "adjoint logic" in which the discretization and codiscretization modalities are characterized using a judgmental formalism of "crisp variables". This yields type theories that we call "spatial" and "cohesive", in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the "fundamental ∞-groupoid" or "shape"), disentangling the "identifications" of Homotopy Type Theory from the "continuous paths" of topology. In a further refinement called "real-cohesion", the shape is determined by continuous maps from the real numbers, as in classical algebraic topology. This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwer's fixed-point theorem.

Change Actions: Models of Generalised Differentiation

“…It is a topic of ongoing research to put this fact in the context of the recent advances in differential cohesive type theory [Gross et al, 2018;Shulman, 2018] (the internal language of a topos with differential cohesion), which have recently been used to give a constructive formalization of Brouwer's fixed point theorem [Shulman, 2018].…”

Univalent foundations as structuralist foundations

“…For example, this is true in the case of categories because an isomorphism between two objects also induces a bijection between corresponding hom-sets. 47 Yet we are perfectly well-able to imagine criteria of identity that do not induce equivalences on the rest of the components 46 Here ¬S(b) can be thought of as notation for S(b) → 0. 47 More precisely, for any other object c an isomorphism f : a ∼ = b induces a bijec-of the structure but for which it is not contradictory to assert a saturation axiom in the style of univalent categories.…”

“…Most prominently, there is a version of HoTT called cohesive HoTT (cf. [41,42,46]) that makes the interaction between the synthetic and the "analytic" side more robust, essentially by adding a kind of "topological" (or "cohesive") structure to bare homotopy types. This extra structure allows one to recover the "analytic" versions of mathematical structures from their synthetic versions.…”

“…In particular, one can prove (cf. [46]) that the realization of the synthetic S 1 is equivalent to the topological circle, thus solving the problem described above. Perhaps this means that an argument could be made that cohesive HoTT -even though motivated by questions in the foundations of higher gauge theories in physics -could actually provide a worthy alternative "foundational" formalization of UF.…”

Constructing Infinitary Quotient-Inductive Types

“…We only establish the computation property up to propositional rather than defini tional equality; so, using the terminology of Shulman[24], these are typal quotient-in ductive types.…”