An (,1)(\infty,1)-functor is a homomorphism between (∞,1)-categories. It generalizes

An (,1)(\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 (,1)(\infty,1)-functors between two (,1)(\infty,1)-categories form an (∞,1)-category of (∞,1)-functors.


The details of the definition depend on the model chosen for (∞,1)-categories.

  1. quasi-category

  2. simplicially enriched category

  3. Segal category

  4. complete Segal space

In terms of quasi-categories


For CC and DD quasi-categories, an (,1)(\infty,1)-functor F:CDF : C \to D is simply a morphism of the underlying simplicial sets.

A natural transformation η:FG\eta : F \to G between two such (,1)(\infty,1)-functors is a simplicial homotopy

C i 0 F C×Δ[1] η D i 1 G C. \array{ C \\ {}^{\mathllap{i_0}}\downarrow & \searrow^{\mathrlap{F}} \\ C \times \Delta[1] &\stackrel{\eta}{\to}& D \\ {}^{\mathllap{i_1}}\uparrow & \nearrow_{G} \\ C } \,.

A modification ρ\rho between natural transformations is an order 2 simplicial homotopy

ρ:C×Δ[2]D. \rho : C \times \Delta[2] \to D \,.

Generally a kk-transfor ϕ\phi of (,1)(\infty,1)-functors is a simplicial homotopy of order kk between the corresponding quasi-categories

ϕ:C×Δ[k]D. \phi : C \times \Delta[k] \to D \,.

In total, the (∞,1)-category of (∞,1)-functors between given quasi-categories CC and DD is the simplicial function complex

(,1)Cat(C,D):=sSet(C,D):= kΔΔ[k]Hom sSet(C×Δ[k],D) (\infty,1)Cat(C,D) := sSet(C,D) := \int^{k \in \Delta} \Delta[k] \cdot Hom_{sSet}(C \times \Delta[k], D)

as computed by the canonical sSet-enrichment of sSetsSet itself.

This serves to define the (∞,1)-category of (∞,1)-functors.


\infty-Pseudo-functors / homotopy presheaves

Let CC be an ordinary category. The above definition in particular serves to generalize the notion of a pseudofunctor (functor up to homotopy)

F:C opGrpd F : C^{op} \to Grpd

with values in the 2-category Grpd as it appears in the theory of stacks/2-sheaves:

let KanCplxsSetKanCplx \subset sSet be the full subcategory of sSet on the Kan complexes. This is naturally a simplicially enriched category. Write N(KanCplx)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)N(C^{op}) for the ordinary nerve of the ordinary category C opC^{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 CC is a morphism of simplicial sets

F:N(C op)N(KanCplx). F : N(C^{op}) \to N(\mathbf{KanCplx}) \,.

One sees easily in low degrees that this does look like the a pseudofunctor there:

  1. the 1-cells of N(C)N(C) are just the morphisms in CC, so that on 1-cells we have that FF is an assignment

    F:(xfy)(F(x)F(f)F(y) F : (x \stackrel{f}{\leftarrow} y) \mapsto (F(x) \stackrel{F(f)}{\to} F(y)

    of morphisms in CC to morphisms in KanCplxKanCplx, as befits a functor;

  2. the 2-cells of N(C)N(C) are pairs of composable morphisms, so that on 2-cells we have that FF is an assignment

    F:( y g f x gf z)( F(y) F(g) F(f,g) F(f) F(x) F(gf) F(z)) F : \left( \array{ && y \\ & {}^{\mathllap{g}}\swarrow & & \nwarrow^{\mathrlap{f}} \\ x &&\stackrel{g \circ f}{\leftarrow}&& z } \right) \;\; \mapsto \;\; \left( \array{ && F(y) \\ & {}^{\mathllap{F(g)}}\nearrow & \Downarrow^{\mathrlap{F(f,g)}} & \searrow^{\mathrlap{F(f)}} \\ F(x) &&\stackrel{F(g \circ f)}{\rightarrow}&& F(z) } \right)

    which means that FF does not necessarily respect the composition of moprhisms, but instead does introduce homotopies F(f,g)F(f,g) for very pairs of composable morphisms, which measure how F(g)F(f)F(g)\circ F(f) differs from F(gf)F(g \circ f). These are precisely the homotopies that one sees also in an ordinary pseudofunctor. But for our (,1)(\infty,1)-functor there are now also higher and higher homotopies:

  3. the 3-cells of N(C)N(C) are triples of composable morphisms (f,g,h)(f,g,h) in CC. They are sent by FF to a tetrahedron that consists of a homotopy-of-homotopies from the F(f,g)F(h,gf)F(f,g) \cdot F( h , g\circ f ) to F(g,h)F(f,hg)F(g, h) \cdot F(f , h \circ g);

  4. and so on.

For more see (∞,1)-presheaf.


It turns out that every (,1)(\infty,1)-functor CGrpdC \to \infty Grpd can be rectified to an ordinary (sSet-enriched) functor with values in Kan complexes.


For C=N(C)C = N(\mathbf{C}) a quasi-category given as the homotopy coherent nerve of a Kan-complex enriched category C\mathbf{C} (which may for instance be just an ordinary 1-category), write

[C op,sSet] [\mathbf{C}^{op}, \mathbf{sSet}]

for the sSet-enriched category of ordinary (sSetsSet-enriched) functors (respecting composition strictly).

Then: every (,1)(\infty,1)-functor N(C op)GrpdN(\mathbf{C}^{op}) \to \infty Grpd is equivalent to a strictly composition respecting functor of this sort. Precisely: write [C op,sSet] [\mathbf{C}^{op}, \mathbf{sSet}]^\circ for the full sSet\mathbf{sSet}-enriched subcategory on those strict functors that are fibrant and cofibrant in a model structure on simplicial presheaves on C\mathbf{C}. Then we have an equivalence of (∞,1)-categories

Hom (,1)Cat(N(C op),Grpd)N([C op,sSet] ). Hom_{(\infty,1)Cat}(N(\mathbf{C}^{op}), \infty Grpd) \simeq N([\mathbf{C}^{op}, \mathbf{sSet}]^\circ) \,.

More on this is at (∞,1)-category of (∞,1)-presheaves.


section 1.2.7 in

discusses morphisms of quasi-categories.

Revised on October 16, 2017 01:38:37 by Mike Shulman (