nLab
indexed (infinity, 1)-category

Contents

Context

(,1)(\infty,1)-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

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

(,1)(\infty,1)-topos theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

An indexed (∞,1)-category is the (∞,1)-category theoretic analogue of an indexed category. If SS is an (∞,1)-category, then an indexed (∞,1)-category is a functor from S opS^{op} to (∞,1)Cat.

The (∞,1)-Grothendieck construction establishes an equivalence between (∞,1)-categories indexed by SS and Cartesian fibrations over SS.

Variants

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 (,1)(\infty, 1)-category can be formed in this way. In particular, exponentiable ∞-functors are treated in (AyalaFrancis by maps to (,1)Prof(\infty,1)Prof, but with extra ‘flagged’ structure. An (,1)(\infty, 1)-version of proarrow equipments should work here.

In addition, for some purposes conditions may be placed on the indexing category, SS, such as the atomic orbital \infty-categories of Parametrized Higher Category Theory and Higher Algebra.

References

Last revised on August 24, 2017 at 03:43:47. See the history of this page for a list of all contributions to it.