nLab
homotopy 2-category

Contents

Context

2-category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The homotopy 2-category of an (∞,n)-category 𝒞\mathcal{C} is the 2-category Ho 2(𝒞)Ho_2(\mathcal{C}) with the same objects and 1-morphisms as 𝒞\mathcal{C} and with the 2-morphisms being the equivalence classes of 2-morphisms of 𝒞\mathcal{C}.

In other words, for every pair X,YX,Y of objects in 𝒞\mathcal{C}, the hom-category Ho 2(𝒞)(X,Y)Ho_2(\mathcal{C})(X,Y) is the ordinary homotopy category of the (,n1)(\infty,n-1)-category 𝒞(X,Y)\mathcal{C}(X,Y).

Examples

References

The homotopy 2-category of topological spaces, continuous functions and homotopies, regarded as a strict (2,1)-category, hence a Grpd-enriched category:

On the Toda bracket understood as homotopy-coherent pasting diagrams in a pointed homotopy 2-category:

In the context of the homotopy 2-category of (∞,1)-categories:

Last revised on August 30, 2021 at 09:31:36. See the history of this page for a list of all contributions to it.