locally small (infinity,1)-category

**(∞,1)-category theory**
## Background
* category theory
* higher category theory
* (n,r)-category
## Basic concepts
* (∞,1)-category
* hom-objects
* equivalences in/of $(\infty,1)$-categories
* sub-(∞,1)-category
* reflective sub-(∞,1)-category
* reflective localization
* opposite (∞,1)-category
* over (∞,1)-category
* join of quasi-categories
* (∞,1)-functor
* exact (∞,1)-functor
* (∞,1)-category of (∞,1)-functors
* (∞,1)-category of (∞,1)-presheaves
* **fibrations**
* inner fibration
* left/right fibration
* Cartesian fibration
* Cartesian morphism
## Universal constructions
* limit
* terminal object
* adjoint functors
## Local presentation
* locally presentable
* essentially small
* locally small
* accessible
* idempotent-complete
## Theorems
* (∞,1)-Yoneda lemma
* (∞,1)-Grothendieck construction
* adjoint (∞,1)-functor theorem
* (∞,1)-monadicity theorem
## Extra stuff, structure, properties
* stable (∞,1)-category
* (∞,1)-topos
## Models
* category with weak equivalences
* model category
* derivator
* quasi-category
* model structure for quasi-categories
* model structure for Cartesian fibrations
* relation to simplicial categories
* homotopy coherent nerve
* simplicial model category
* presentable quasi-category
* Kan complex
* model structure for Kan complexes

The notion of *locally small $(\infty,1)$-category* is the generalization of the notion of locally small category from category theory to (∞,1)-category theory.

A quasi-category $C$ is **locally small** if for all objects $x,y \in C$ the hom ∞-groupoid $Hom_C(x,y)$ is essentially small.

This appears as HTT, below prop. 5.4.1.7.

A quasi-category $C$ is locally small precisely if the following equivalent condition holds:

for every small set $S$ of objects in $C$, the full sub-quasi-category on $S$ is essentially small.

This is the topic of section 5.4.1 of

Created on April 14, 2010 18:48:38
by Urs Schreiber
(131.211.232.49)