**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**

- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

Where the homotopy hypothesis is the assertion that ∞-groupoids are equivalent to topological spaces (considered modulo weak homotopy equivalence), in AFR15 the authors construct an equivalence for (∞,1)-categories. They do so in terms of what are called **striation sheaves**, which are sheaves on conically smooth stratified spaces satisfying a certain descent condition.

The construction relies on a fully faithful embedding of conically smooth stratified spaces, and conically smooth maps among them, into $(\infty, 1)$-categories via the exit-path functor, which maps stratified spaces to paths within them that once they leave a stratum do not re-enter it.

- David Ayala, John Francis, Nick Rozenblyum,
*A stratified homotopy hypothesis*, arXiv

Created on October 27, 2017 at 05:44:51. See the history of this page for a list of all contributions to it.