An -functor is a homomorphism between (∞,1)-categories. It generalizes
An -functor is functorial (respects composition) only up to coherent higher homotopies. It may be thought of as a homotopy coherent functor or strongly homotopy functor.
The collection of all -functors between two -categories form an (∞,1)-category of (∞,1)-functors.
The details of the definition depend on the model chosen for (∞,1)-categories.
simplicially enriched category
complete Segal space
In terms of quasi-categories
For and quasi-categories, an -functor is simply a morphism of the underlying simplicial sets.
A natural transformation between two such -functors is a simplicial homotopy
A modification between natural transformations is an order 2 simplicial homotopy
Generally a -transfor of -functors is a simplicial homotopy of order between the corresponding quasi-categories
In total, the (∞,1)-category of (∞,1)-functors between given quasi-categories and is the simplicial function complex
as computed by the canonical sSet-enrichment of itself.
This serves to define the (∞,1)-category of (∞,1)-functors.
-Pseudo-functors / homotopy presheaves
Let be an ordinary category. The above definition in particular serves to generalize the notion of a pseudofunctor (functor up to homotopy)
with values in the 2-category Grpd as it appears in the theory of stacks/2-sheaves:
let be the full subcategory of sSet on the Kan complexes. This is naturally a simplicially enriched category. Write for the homotopy coherent nerve of this simplicially enriched category. This is the quasi-category-incarnaton of ∞Grpd.
Write for the ordinary nerve of the ordinary category (passing to the opposite category is just a convention here, with no effect on the substance of the statement). Then an -pseudofunctor or (∞,1)-presheaf or homotopy presheaf on is a morphism of simplicial sets
One sees easily in low degrees that this does look like the a pseudofunctor there:
the 1-cells of are just the morphisms in , so that on 1-cells we have that is an assignment
of morphisms in to morphisms in , as befits a functor;
the 2-cells of are pairs of composable morphisms, so that on 2-cells we have that is an assignment
which means that does not necessarily respect the composition of moprhisms, but instead does introduce homotopies for very pairs of composable morphisms, which measure how differs from . These are precisely the homotopies that one sees also in an ordinary pseudofunctor. But for our -functor there are now also higher and higher homotopies:
the 3-cells of are triples of composable morphisms in . They are sent by to a tetrahedron that consists of a homotopy-of-homotopies from the to ;
and so on.
For more see (∞,1)-presheaf.
It turns out that every -functor can be rectified to an ordinary (sSet-enriched) functor with values in Kan complexes.
For a quasi-catwgeory given as the homotopy coherent nerve of a Kan-complex enriched category (which may for instance be just an ordinary 1-category), write
for the sSet-enriched category of ordinary (-enriched) functors (respecting composition strictly).
Then: every -functor is equivalent to a strictly composition respecting functor of this sort. Precisely: write for the full -enriched subcategory on those strict functors that are fibrant and cofibrant in the model structure on simplicial presheaves on . Then we have an equivalence of ∞-groupoids
More on this is at (∞,1)-category of (∞,1)-presheaves.
section 1.2.7 in
discusses morphisms of quasi-categories.