# nLab (infinity,1)-functor

Contents

### Context

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# Contents

## Idea

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

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.

## 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 $(\infty,1)$-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

$\array{ C \\ {}^{\mathllap{i_0}}\downarrow & \searrow^{\mathrlap{F}} \\ C \times \Delta &\stackrel{\eta}{\to}& D \\ {}^{\mathllap{i_1}}\uparrow & \nearrow_{G} \\ C } \,.$

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

$\rho : C \times \Delta \to D \,.$

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

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

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

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

as computed by the canonical sSet-enrichment of $sSet$ 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)

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

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

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

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

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

of morphisms in $C$ to morphisms in $KanCplx$, 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

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

which means that $F$ does not necessarily respect the composition of morphisms, 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:

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 $(\infty,1)$-functor $C \to \infty Grpd$ can be rectified to an ordinary (sSet-enriched) functor with values in Kan complexes.

###### Theorem

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

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

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

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

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

section 1.2.7 in

discusses morphisms of quasi-categories.