A Cubical Approach to Synthetic Homotopy Theory

Abstract

Abstract-Homotopy theory can be developed synthetically in homotopy type theory, using types to describe spaces, the identity type to describe paths in a space, and iterated identity types to describe higher-dimensional paths. While some aspects of homotopy theory have been developed synthetically and formalized in proof assistants, some seemingly easy examples have proved difficult because the required manipulations of paths becomes complicated. In this paper, we describe a cubical approach to developing homotopy theory within type theory. The identity type is complemented with higher-dimensional cube types, such as a type of squares, dependent on four points and four lines, and a type of three-dimensional cubes, dependent on the boundary of a cube. Path-over-a-path types and higher generalizations are used to describe cubes in a fibration over a cube in the base. These higher-dimensional cube and path-over types can be defined from the usual identity type, but isolating them as independent conceptual abstractions has allowed for the formalization of some previously difficult examples.

Homotopy Type Theory in Lean

“…Likewise, we can prove the functoriality of functions with respect to equality: For a function f : A → B and p : a = a we define ap f p : f a = f a by induction on p. Using an equality p : a = a in a type A to compare elements of two fibers in a type family C over A, we define the transport of an element x : C a along p as p x : C a . For higher paths and dependent paths, we follow what Dan Licata calls the "cubical approach" [16]. The basic notion is that of pathover, or a "path over a path", which compares elements x : C a and y : C a in different fibers of a type family over some path p : a = a in the base type.…”

From Reversible Programs to Univalent Universes and Back

“…As illustrated in the figure below, the point transport P p u is in the space P y. A path from that point to another point v in P y can be viewed as a virtual "path" between u and v that "lies over" p. Following Licata and Brunerie [24], we often use the syntax u == v [ P Ó p ] for the path transport P p u == v to reinforce this perspective. In other words, the curved "path" between u and v below consists of first transporting u to the space P y along p and then following the straight path in P y to v: Given a fibration P and points x, y, u, and v as above, we have the following characterization of dependent paths in the total space:…”

The James Construction and $$\pi _4(\mathbb {S}^{3})$$ in Homotopy Type Theory

“…where α : A × B → north = ΣA south is defined by α(x, y) := merid(x), and we use the 3 × 3lemma (cf section VII of [5]) which states that the pushout of the pushouts of the rows is equivalent to the pushout of the pushouts of the columns. The pushout of the top row is equivalent to ΣA ∨ ΣB, the pushout of the middle row is equivalent to the join A * B and the pushout of the bottom row is contractible, so the pushout of the pushouts of the rows is equivalent to 1 ⊔ A * B (ΣA ∨ ΣB) for the map W A,B : A * B → ΣA ∨ ΣB defined by…”

“…The idea is that we start with the disjoint sum A + B, and for every element c of C we add a new path from inl( f (c)) to inr(g(c)). The induction principle states that, given a dependent type P : A ⊔ C B → Type, we can define a function h : We are using here the notion of dependent paths (see [5]): given a type X, a dependent type P : X → Type, a path p : x = x ′ in X and two points u : P(x) and v : P(x ′ ), the type u = P p v represents paths in P going from u to v and lying over p. Given h : (x : X) → Q(x) and q : x = X x ′ , the term apd h (q) is the application of h to q, which is a dependent path in Q, over q, and from h(x) to h(x ′ ).…”