equivalences in/of $(\infty,1)$-categories
An $(\infty,1)$-functor is a homomorphism between (∞,1)-categories. It generalizes
the notion of functor between categories;
the notion of pseudofunctor from a category to a (2,1)-category;
the notion of 2-functor between (2,1)-categories;
the notion of n-functor between (n,1)-categories.
An $(\infty,1)$-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 $(\infty,1)$-functors between two $(\infty,1)$-categories form an (∞,1)-category of (∞,1)-functors.
The details of the definition depend on the model chosen for (∞,1)-categories.
For $C$ and $D$ quasi-categories, an $(\infty,1)$-functor $F : C \to D$ is simply a morphism of the underlying simplicial sets.
A natural transformation $\eta : F \to G$ between two such $(\infty,1)$-functors is a simplicial homotopy
A modification $\rho$ between natural transformations is an order 2 simplicial homotopy
Generally a $k$-transfor $\phi$ of $(\infty,1)$-functors is a simplicial homotopy of order $k$ between the corresponding quasi-categories
In total, the (∞,1)-category of (∞,1)-functors between given quasi-categories $C$ and $D$ is the simplicial function complex
as computed by the canonical sSet-enrichment of $sSet$ itself.
This serves to define the (∞,1)-category of (∞,1)-functors.
Let $C$ 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 $KanCplx \subset sSet$ be the full subcategory of sSet on the Kan complexes. This is naturally a simplicially enriched category. Write $N(\mathbf{KanCplx})$ for the homotopy coherent nerve of this simplicially enriched category. This is the quasi-category-incarnaton of ∞Grpd.
Write $N(C^{op})$ for the ordinary nerve of the ordinary category $C^{op}$ (passing to the opposite category is just a convention here, with no effect on the substance of the statement). Then an $\infty$-pseudofunctor or (∞,1)-presheaf or homotopy presheaf on $C$ 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 $N(C)$ are just the morphisms in $C$, so that on 1-cells we have that $F$ is an assignment
of morphisms in $C$ to morphisms in $KanCplx$, as befits a functor;
the 2-cells of $N(C)$ are pairs of composable morphisms, so that on 2-cells we have that $F$ is an assignment
which means that $F$ does not necessarily respect the composition of moprhisms, but instead does introduce homotopies $F(f,g)$ for very pairs of composable morphisms, which measure how $F(g)\circ F(f)$ differs from $F(g \circ f)$. These are precisely the homotopies that one sees also in an ordinary pseudofunctor. But for our $(\infty,1)$-functor there are now also higher and higher homotopies:
the 3-cells of $N(C)$ are triples of composable morphisms $(f,g,h)$ in $C$. They are sent by $F$ to a tetrahedron that consists of a homotopy-of-homotopies from the $F(f,g) \cdot F( h , g\circ f )$ to $F(g, h) \cdot F(f , h \circ g)$;
and so on.
For more see (∞,1)-presheaf.
It turns out that every $(\infty,1)$-functor $C \to \infty Grpd$ can be rectified to an ordinary (sSet-enriched) functor with values in Kan complexes.
For $C = N(\mathbf{C})$ a quasi-category given as the homotopy coherent nerve of a Kan-complex enriched category $\mathbf{C}$ (which may for instance be just an ordinary 1-category), write
for the sSet-enriched category of ordinary ($sSet$-enriched) functors (respecting composition strictly).
Then: every $(\infty,1)$-functor $N(\mathbf{C}^{op}) \to \infty Grpd$ is equivalent to a strictly composition respecting functor of this sort. Precisely: write $[\mathbf{C}^{op}, \mathbf{sSet}]^\circ$ for the full $\mathbf{sSet}$-enriched subcategory on those strict functors that are fibrant and cofibrant in a model structure on simplicial presheaves on $\mathbf{C}$. Then we have an equivalence of (∞,1)-categories
More on this is at (∞,1)-category of (∞,1)-presheaves.
(∞,1)-functor
section 1.2.7 in
discusses morphisms of quasi-categories.