Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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
stabilization hypothesis-theorem
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An indexed (∞,1)-category is the (∞,1)-category theoretic analogue of an indexed category. If is an (∞,1)-category, then an indexed (∞,1)-category is a functor from to (∞,1)Cat.
The (∞,1)-Grothendieck construction establishes an equivalence between (∞,1)-categories indexed by and Cartesian fibrations over .
Varieties of indexed (∞,1)-category can be formed by requiring that the target be monoidal, symmetric, closed, and so on.
Not all kinds of fibration of -category can be formed in this way. In particular, exponentiable ∞-functors are treated in (AyalaFrancis by maps to , but with extra ‘flagged’ structure. An -version of proarrow equipments should work here.
In addition, for some purposes conditions may be placed on the indexing category, , such as the atomic orbital -categories of Parametrized Higher Category Theory and Higher Algebra.
Last revised on August 24, 2017 at 07:43:47. See the history of this page for a list of all contributions to it.