We introduce a dependent type theory whose models are weak ω-categories, generalizing Brunerie's definition of ω-groupoids. Our type theory is based on the definition of ω-categories given by Maltsiniotis, himself inspired by Grothendieck's approach to the definition of ω-groupoids. In this setup, ω-categories are defined as presheaves preserving globular colimits over a certain category, called a coherator. The coherator encodes all operations required to be present in an ω-category: both the compositions of pasting schemes as well as their coherences. Our main contribution is to provide a canonical type-theoretical characterization of pasting schemes as contexts which can be derived from inference rules. Finally, we present an implementation of a corresponding proof system.
Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs of homotopy-theoretic theorems, in a way that is very amenable to computer-checked proofs in proof assistants such as Coq and Agda. In this paper, we give a computer-checked construction of Eilenberg-MacLane spaces. For an abelian group G, an EilenbergMacLane space K(G, n) is a space (type) whose n th homotopy group is G, and whose homotopy groups are trivial otherwise. These spaces are a basic tool in algebraic topology; for example, they can be used to build spaces with specified homotopy groups, and to define the notion of cohomology with coefficients in G. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far.
This paper continues investigations in "synthetic homotopy theory": the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory.We present a mechanized proof of the Blakers-Massey connectivity theorem, a result relating the higher-dimensional homotopy groups of a pushout type (roughly, a space constructed by gluing two spaces along a shared subspace) to those of the components of the pushout. This theorem gives important information about the pushout type, and has a number of useful corollaries, including the Freudenthal suspension theorem, which has been studied in previous formalizations.The new proof is more elementary than existing ones in abstract homotopy-theoretic settings, and the mechanization is concise and high-level, thanks to novel combinations of ideas from homotopy theory and type theory.
We study the dependent type theory CaTT, introduced by Finster and Mimram, which presents the theory of weak ω-categories, following the idea that type theories can be considered as presentations of generalized algebraic theories. Our main contribution is a formal proof that the models of this type theory correspond precisely to weak ω-categories, as defined by Maltsiniotis, by generalizing a definition proposed by Grothendieck for weak ωgroupoids: Those are defined as suitable presheaves over a cat-coherator, which is a category encoding structure expected to be found in an ω-category. This comparison is established by proving the initiality conjecture for the type theory CaTT, in a way which suggests the possible generalization to a nerve theorem for a certain class of dependent type theories
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