nLab
(infinity,1)-functor

Idea

An $(\mathrm{\infty },1)$-functor is a morphism 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 $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-functors between two $(\mathrm{\infty },1)$-categories form an (∞,1)-category of (∞,1)-functors.

Definition

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

Definition

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

A natural transformation $\eta :F\to G$ between two such $(\mathrm{\infty },1)$-functors is a simplicial homotopy

$$\begin{array}{c}C\\ {}^{i_0}\downarrow & {\searrow }^F\\ C\times \Delta [1]& \stackrel{\eta }{\to }& D\\ {}^{i_1}\uparrow & {\nearrow }_G\\ C\end{array}.$$\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 $\rho$ between natural transformations is an order 2 simplicial homotopy

$$\rho :C\times \Delta [2]\to D.$$\rho : C \times \Delta[2] \to D \,.

Generally a $k$-transfor $\varphi$ of $(\mathrm{\infty },1)$-functors is a simplicial homotopy of order $k$ between the corresponding quasi-categories

$$\varphi :C\times \Delta [k]\to D.$$\phi : C \times \Delta[k] \to D \,.

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

$$(\mathrm{\infty },1)\mathrm{Cat}(C,D):=\mathrm{sSet}(C,D):={\int }^{k\in \Delta }\Delta [k]\cdot {\mathrm{Hom}}_{\mathrm{sSet}}(C\times \Delta [k],D)$$(\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 $\mathrm{sSet}$ itself.

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

Examples

$\mathrm{\infty }$-Pseudo-functors / homotopy presheaves

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

$$F:C^{\mathrm{op}}\to \mathrm{Grpd}$$F : C^{op} \to Grpd

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

let $\mathrm{KanCplx}\subset \mathrm{sSet}$ be the full subcategory of sSet on the Kan complexes. This is naturally a simplicially enriched category. Write $N(\mathrm{KanCplx})$ for the homotopy coherent nerve of this simplicially enriched category. This is the quasi-category-incarnaton of ∞Grpd.

Write $N(C^{\mathrm{op}})$ for the ordinary nerve of the ordinary category $C^{\mathrm{op}}$ (passing to the opposite category is just a convention here, with no effect on the substance of the statement). Then an $\mathrm{\infty }$-pseudofunctor or (∞,1)-presheaf or homotopy presheaf on $C$ is a morphism of simplicial sets

$$F:N(C^{\mathrm{op}})\to N(\mathrm{KanCplx}).$$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 $N(C)$ are just the morphisms in $C$, so that on 1-cells we have that $F$ is an assignment

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

   of morphisms in $C$ to morphisms in $\mathrm{KanCplx}$, as befits a functor;

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

   $$F:\left(\begin{array}{ccc}& & y\\ & {}^g\swarrow & & {\nwarrow }^f\\ x& & \stackrel{g\circ f}{\leftarrow }& & z\end{array}\right)\mapsto \left(\begin{array}{ccc}& & F(y)\\ & {}^{F(g)}\nearrow & {\Downarrow }^{F(f,g)}& {\searrow }^{F(f)}\\ F(x)& & \stackrel{F(g\circ f)}{\to }& & F(z)\end{array}\right)$$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 $F$ 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 $(\mathrm{\infty },1)$-functor there are now also higher and higher homotopies:

3. 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 $F(g,h)\cdot F(f,h\circ g)$;

4. and so on.

For more see (∞,1)-presheaf.

Properties

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

Theorem

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

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

for the sSet-enriched category of ordinary ($\mathrm{sSet}$-enriched) functors (respecting composition strictly).

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

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

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

Related concepts

• functor

• 2-functor / pseudo-functor

• n-functor

• $(\mathrm{\infty },1)$-functor

References

section 1.2.7 in

• Jacob Lurie, Higher Topos Theory

discusses morphisms of quasi-categories.