# nLab (infinity,1)-functor

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

## Idea

An $\left(\infty ,1\right)$-functor is a morphism between (∞,1)-categories. It generalizes

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

## Definition

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

### In terms of quasi-categories

###### Definition

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

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

$\begin{array}{c}C\\ {}^{{i}_{0}}↓& {↘}^{F}\\ C×\Delta \left[1\right]& \stackrel{\eta }{\to }& D\\ {}^{{i}_{1}}↑& {↗}_{G}\\ C\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\rho :C×\Delta \left[2\right]\to D\phantom{\rule{thinmathspace}{0ex}}.$\rho : C \times \Delta[2] \to D \,.

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

$\varphi :C×\Delta \left[k\right]\to D\phantom{\rule{thinmathspace}{0ex}}.$\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

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

### $\infty$-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 $N\left(\mathrm{KanCplx}\right)$ for the homotopy coherent nerve of this simplicially enriched category. This is the quasi-category-incarnaton of ∞Grpd.

Write $N\left({C}^{\mathrm{op}}\right)$ 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 $\infty$-pseudofunctor or (∞,1)-presheaf or homotopy presheaf on $C$ is a morphism of simplicial sets

$F:N\left({C}^{\mathrm{op}}\right)\to N\left(\mathrm{KanCplx}\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $N\left(C\right)$ are just the morphisms in $C$, so that on 1-cells we have that $F$ is an assignment

$F:\left(x\stackrel{f}{←}y\right)↦\left(F\left(x\right)\stackrel{F\left(f\right)}{\to }F\left(y\right)$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 $N\left(C\right)$ are pairs of composable morphisms, so that on 2-cells we have that $F$ is an assignment

$F:\left(\begin{array}{ccc}& & y\\ & {}^{g}↙& & {↖}^{f}\\ x& & \stackrel{g\circ f}{←}& & z\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}& & F\left(y\right)\\ & {}^{F\left(g\right)}↗& {⇓}^{F\left(f,g\right)}& {↘}^{F\left(f\right)}\\ F\left(x\right)& & \stackrel{F\left(g\circ f\right)}{\to }& & F\left(z\right)\end{array}\right)$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 $F\left(f,g\right)$ for very pairs of composable morphisms, which measure how $F\left(g\right)\circ F\left(f\right)$ differs from $F\left(g\circ f\right)$. These are precisely the homotopies that one sees also in an ordinary pseudofunctor. But for our $\left(\infty ,1\right)$-functor there are now also higher and higher homotopies:

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

4. and so on.

For more see (∞,1)-presheaf.

## Properties

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

###### Theorem

For $C=N\left(C\right)$ 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

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

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

Then: every $\left(\infty ,1\right)$-functor $N\left({C}^{\mathrm{op}}\right)\to \infty \mathrm{Grpd}$ is equivalent to a strictly composition respecting functor of this sort. Precisely: write $\left[{C}^{\mathrm{op}},\mathrm{KanCplx}{\right]}^{\circ }$ 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

$\left(\infty ,1\right)\mathrm{Cat}\left(N\left({C}^{\mathrm{op}}\right),\infty \mathrm{Grpd}\right)\simeq \left[{C}^{\mathrm{op}},\mathrm{sSet}{\right]}^{\circ }\phantom{\rule{thinmathspace}{0ex}}.$(\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.

## References

section 1.2.7 in

discusses morphisms of quasi-categories.

Revised on February 5, 2013 02:16:36 by Urs Schreiber (89.204.154.134)