Homotopy limits in type theory

Abstract

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.Comment: 33 pages; v3: theorem numbering changed since v2 to match journal versio

Quasicategories of Frames of Cofibration Categories

“…Every categorical model of type theory is known to carry the structure of a fibration category [AKL15] and, by our results, its simplicial localization can be realized as the quasicategory of frames. This realization proved convenient for the purpose of solving the problem in question.…”

Locally cartesian closed quasi‐categories from type theory

“…The following theorem justifies (at least partially) our interest in fibration categories: (The construction in [1] uses the assumption that every dependent projection is isomorphic to a basic one. This always holds in C cxt ; so we may apply the construction there, and then transfer the fibration category structure back to C along the canonical equivalence C → C cxt .…”

“…Let E [1] denote the nerve of the contractible groupoid on two objects. Two maps f, g : C → D between quasi-categories are E[1]-homotopic (notation: f ∼ E [1] g) if there exists a map H : The join of simplicial sets is a unique functor : sSet × sSet → sSet with natural transfor-…”

“…We will construct a 1-simplex X : D [1] op → C in A ↓ N f G (thus, we must have X|D[0] = A). Since both F and G are exact, we put X 1 k := GF A 0 k .…”

Constructions with Non-Recursive Higher Inductive Types

“…A systematic treatment of homotopy limits over graphs has been presented by (Avigad et al 2015), which comes with a rigorous formalisation in Coq.…”