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1.4.1 Examples of Functors

Let us begin by illustrating Definition 1.4.0.1 in some special cases.

Example 1.4.1.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories. It follows from Proposition 1.2.2.1 that the formation of nerves induces a bijection

\[ \xymatrix { \{ \text{Functors of $\infty $-categories from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$} \} \ar [d]^{\sim } \\ \{ \text{Functors of ordinary categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$} \} . } \]

In other words, Definition 1.4.0.1 can be regarded as a generalization of the usual notion of functor to the setting of $\infty $-categories.

Example 1.4.1.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{D}}$ be an ordinary category. Using Proposition 1.3.6.2, we obtain a bijection

\[ \xymatrix { \{ \text{Functors of $\infty $-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$} \} \ar [d]^{\sim } \\ \{ \text{Functors of ordinary categories from $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to $\operatorname{\mathcal{D}}$}\} . } \]

Remark 1.4.1.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:

$(a)$

To each object $X \in \operatorname{\mathcal{C}}$ the functor $F$ assigns an object of $\operatorname{\mathcal{D}}$, which we will denote by $F(X)$ (or sometimes more simply by $FX$).

$(b)$

To each morphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$, the functor $F$ assigns a morphism $F(f): F(X) \rightarrow F(Y)$ in the $\infty $-category $\operatorname{\mathcal{D}}$.

$(c)$

For every object $X \in \operatorname{\mathcal{C}}$, the functor $F$ carries the identity morphism $\operatorname{id}_{X}: X \rightarrow X$ in $\operatorname{\mathcal{C}}$ to the identity morphism $\operatorname{id}_{F(X)}: F(X) \rightarrow F(X)$ in $\operatorname{\mathcal{D}}$.

$(d)$

If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are morphisms in $\operatorname{\mathcal{C}}$ and $h$ is a composition of $f$ and $g$ (in the sense of Definition 1.3.4.1), then the morphism $F(h): F(X) \rightarrow F(Z)$ is a composition of $F(f)$ and $F(g)$.

Warning 1.4.1.4. To define a functor $F$ from an ordinary category $\operatorname{\mathcal{C}}$ to an ordinary category $\operatorname{\mathcal{D}}$, it suffices to specify the values of $F$ on objects and morphisms (as described in $(a)$ and $(b)$ of Remark 1.4.1.3) and to verify that $F$ is compatible with the formation of composition and identity morphisms (as described in $(c)$ and $(d)$ of Remark 1.4.1.3). In the $\infty $-categorical setting, this is not enough: to give a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, one must specify its values on simplices of all dimensions. Roughly speaking, these values encode the requirement that $F$ is compatible with composition “up to coherent homotopy.” For example, suppose that we are given objects $X,Y,Z \in \operatorname{\mathcal{C}}$ and morphisms $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$. Part $(d)$ of Remark 1.4.1.3 asserts that if $h$ is a composition of $f$ and $g$, then $F(h)$ is a composition of $F(f)$ and $F(g)$. However, we can say more: if $\sigma $ is a $2$-simplex of $\operatorname{\mathcal{C}}$ which witnesses $h$ as a composition of $f$ and $g$, then $F(\sigma )$ is a $2$-simplex of $\operatorname{\mathcal{D}}$ which witnesses $F(h)$ as a composition of $F(f)$ and $F(g)$.

Remark 1.4.1.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. If $f,g: X \rightarrow Y$ are homotopic morphisms of $\operatorname{\mathcal{C}}$, then $F(f), F(g): F(X) \rightarrow F(Y)$ are homotopic morphisms of $\operatorname{\mathcal{D}}$. More precisely, the functor $F$ carries homotopies from $f$ to $g$ (viewed as $2$-simplices of $\operatorname{\mathcal{C}}$) to homotopies from $F(f)$ to $F(g)$ (viewed as $2$-simplices of $\operatorname{\mathcal{D}}$).

Remark 1.4.1.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $f: X \rightarrow Y$ is a morphism in $\operatorname{\mathcal{C}}$ and $g: Y \rightarrow X$ is a homotopy inverse to $f$, then $F(g)$ is a homotopy inverse to $F(f)$. In particular, if $f$ is an equivalence in $\operatorname{\mathcal{C}}$, then $F(f)$ is also an equivalence in $\operatorname{\mathcal{D}}$.

Example 1.4.1.7. Let $X$ be a topological space and let $\operatorname{\mathcal{C}}$ be an ordinary category. To specify a functor of $\infty $-categories $F: \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, one must give a rule which assigns to each continuous map $\sigma : | \Delta ^ n | \rightarrow X$ (viewed as an $n$-simplex of $\operatorname{Sing}_{\bullet }(X)$) a diagram $F(\sigma ) = ( C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow {f_ n} C_ n)$. In particular:

$(a)$

To each point $x \in X$, the functor $F$ assigns an object $F(x) \in \operatorname{\mathcal{C}}$.

$(b)$

To each continuous path $f: [0,1] \rightarrow X$ starting at the point $x = f(0)$ and ending at the point $y = f(1)$, the functor $F$ assigns a morphism $F(f): F(x) \rightarrow F(y)$ in the category $\operatorname{\mathcal{C}}$. The morphism $F(f)$ is automatically an isomorphism (by virtue of Proposition 1.3.7.12 and Remark 1.4.1.6).

$(c)$

For each continuous map $\sigma : | \Delta ^2 | \rightarrow X$ with boundary behavior as depicted in the diagram

\[ \xymatrix { & y \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z, } \]

we have an identity $F(h) = F(g) \circ F(f)$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(x), F(z) )$.

The data of a collection of objects $\{ F(x) \} _{x \in X}$ and isomorphisms $\{ F(f) \} _{ f: [0,1] \rightarrow X}$ satisfying $(c)$ is called a $\operatorname{\mathcal{C}}$-valued local system on $X$. The preceding discussion determines a bijection

\[ \xymatrix { \{ \text{ Functors of $\infty $-categories from $\operatorname{Sing}_{\bullet }(X)$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$} \} \ar [d]^{\sim } \\ \{ \text{ $\operatorname{\mathcal{C}}$-valued local systems on $X$} \} . } \]

By virtue of Example 1.4.1.2, we can also identify local systems with functors from the fundamental groupoid $\pi _{\leq 1}(X)$ into $\operatorname{\mathcal{C}}$.

Remark 1.4.1.8. Let $X$ be a topological space and let $\operatorname{\mathcal{C}}$ be an arbitrary $\infty $-category. Motivated by Example 1.4.1.7, one can define a $\operatorname{\mathcal{C}}$-valued local system on $X$ to be a functor of $\infty $-categories $\operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{\mathcal{C}}$. Beware that this notion generally cannot be reformulated in terms of the fundamental groupoid $\pi _{\leq 1}(X)$.

Example 1.4.1.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be a topological space. Then we have a canonical bijection

\[ \xymatrix { \{ \text{Functors of $\infty $-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{Sing}_{\bullet }(X)$} \} \ar [d]^{\sim } \\ \{ \text{Maps of topological spaces from $| \operatorname{\mathcal{C}}|$ to $X$} \} . } \]

Here $| \operatorname{\mathcal{C}}|$ denotes the geometric realization of the simplicial set $\operatorname{\mathcal{C}}$ (see Definition 1.1.6.1). Beware that neither side has an obvious interpretation in terms of functors between ordinary categories (even in the special case where $\operatorname{\mathcal{C}}$ is the nerve of a category).