* category theory
* higher category theory
## Basic concepts
* equivalences in
* reflective sub-(∞,1)-category
* reflective localization
* opposite (∞,1)-category
* over (∞,1)-category
* join of quasi-categories
* exact (∞,1)-functor
* (∞,1)-category of (∞,1)-functors
* (∞,1)-category of (∞,1)-presheaves
* inner fibration
* left/right fibration
* Cartesian fibration
* Cartesian morphism
## Universal constructions
* terminal object
* adjoint functors
## Local presentation
* locally presentable
* essentially small
* locally small
* (∞,1)-Yoneda lemma
* (∞,1)-Grothendieck construction
* adjoint (∞,1)-functor theorem
* (∞,1)-monadicity theorem
## Extra stuff, structure, properties
* stable (∞,1)-category
* category with weak equivalences
* model 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
Between any two objects in an (∞,1)-category there is an ∞-groupoid of morphisms. It is well-defined up to equivalence. When the -category is incarnated as a quasi-category, there are several explicit ways to extract representatives of this hom-object.
Let be a quasi-category (or any simplicial set), and any two objects. Then write
This defines as an equivalence class of -groupoids, but at the same time defines a particular representative: if is a quasi-category then is a Kan complex that represents this class.
This is useful for many purposes, but is usually hard to compute explicitly. The following three other definitions of representatives of are often useful.
For and as above, write
for the pullback
in sSet of the path object (the cartesian internal hom in sSet with the 1-simplex ) .
for the simplicial set whose -simplices are defined to be those morphisms such that the restriction to is the constant map to and the restriction to is the map to .
for the simplicial set whose -simplices are morphisms which restrict to on and are constant on when restricted to .
Remark The 1-cells in , and are 2-globes in . The 2-cells are commuting squares of vertical composites of 2-globes forming a 3-globe.
Equivalently this may be understood in terms of fibers of over quasi-categories.
Recall that for a morphism, we have the over quasi-category defined by
where on the right we have the set of morphisms in out of the join of simplicial sets that restrict on to .
This comes with the canonical projection , which sends to the restriction .
There is also the other, equivalent, definition of over quasi-category, defined using the other, equivalent, definition of join of quasi-categories by
where on the right we have the pullback in sSet in the diagram
and the equality sign denotes an isomorphism in sSet.
And we have
Proof For the first statement use the identification of with the join of simplicial sets , as described there.
For the second statement use that is the colimit in
Remark One advantage of the representative of is that, by the fact that is an sSet-enriched category, there is a strict composition operation
This is not available for the and
If the simplicial set is a quasi-category, then is a Kan complex.
This is HTT, prop 220.127.116.11.
From the definition it is clear that has fillers for all inner and right outer horns , because these yield inner horns in . The claim follows then with the fact that every right fibration over the point is a Kan complex, as described there.
If is a quasi-category then the canonical inclusions
are homotopy equivalences of Kan complexes.
This is HTT, cor. 18.104.22.168.
As described at join of quasi-categories the canonical morphism is an equivalence of quasi-categories. So for the statement for it suffices to show that this induces an equivalence of fibers over . This follows from the fact that both and are Cartesian fibrations.
See Cartesian fibrations for these statements. This is HTT, prop. 22.214.171.124. (2).
The statement for follows dually.