nLab (infinity,1)-functor

Idea

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.

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 $(\infty,1)$-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

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

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

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

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.

Related concepts

• function

• functor

• 2-functor / pseudofunctor / (2,1)-functor

  • monoidal functor

• n-functor

• (∞,1)-functor

  • monoidal (∞,1)-functor

  • adjoint (∞,1)-functor, (∞,1)-monad

• (∞,n)-functor

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

References

section 1.2.7 in

• Jacob Lurie, Higher Topos Theory

discusses morphisms of quasi-categories.