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2.4.1 Simplicial Enrichment

Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets (Definition 1.1.1.12). Then $\operatorname{Set_{\Delta }}$ admits Cartesian products (Remark 1.1.1.13), and can therefore be endowed with the Cartesian monoidal structure described in Example 2.1.3.2. We will use the term simplicial category to refer to a category which is enriched over $\operatorname{Set_{\Delta }}$, in the sense of Definition 2.1.7.1. For the reader's convenience, we spell this definition out in detail (and establish some notation we will use when discussing simplicial categories, which differs somewhat from the general conventions of §2.1.7).

Definition 2.4.1.1 (Simplicial Categories). A simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ consists of the following data:

$(1)$

A collection $\operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, whose elements we refer to as objects of $\operatorname{\mathcal{C}}_{\bullet }$. We will often abuse notation by writing $X \in \operatorname{\mathcal{C}}_{\bullet }$ to indicate that $X$ is an element of $\operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$.

$(2)$

For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, a simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.

$(3)$

For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, a morphism of simplicial sets

\[ c_{Z,Y,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z)_{\bullet }, \]

which we will refer to as the composition law.

$(4)$

For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a vertex $\operatorname{id}_{X} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{0}$, which we will refer to as the identity morphism of $X$.

These data are required to satisfy the following conditions:

$(A)$

For every quadruple of objects $W,X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the diagram of simplicial sets

\[ \xymatrix@C =-80pt{ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{\bullet } \ar [dl]_{ \operatorname{id}\times c_{Y,X,W} } \ar [dr]^{ c_{Z,Y,X} \times \operatorname{id}} & \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)_{\bullet } \ar [dr]_{ c_{Z,Y,W} } & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{\bullet } \ar [dl]^{ c_{Z,X,W} } \\ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z)_{\bullet } & } \]

commutes (in other words, the composition law of $(3)$ is associative).

$(U)$

For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the maps of simplicial sets

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \{ \operatorname{id}_ X \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet } \xrightarrow { c_{Y,X,X} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \]
\[ \{ \operatorname{id}_ Y \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \xrightarrow { c_{Y,Y,X} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \]

coincide with the projection maps onto the factor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.

Warning 2.4.1.2. The terminology of Definition 2.4.1.1 is not standard. Many authors use the term simplicial category to mean a simplicial object of the category $\operatorname{Cat}$, and the term simplicially enriched category to mean a category enriched over simplicial sets. These notions are closely related: see Remark 2.4.1.12.

Construction 2.4.1.3. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. For every nonnegative integer $n \geq 0$, we define an ordinary category $\operatorname{\mathcal{C}}_{n}$ as follows:

  • The objects of $\operatorname{\mathcal{C}}_ n$ are the objects of $\operatorname{\mathcal{C}}_{\bullet }$.

  • Let $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}_ n) = \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$ be objects of $\operatorname{\mathcal{C}}_ n$. A morphism from $X$ to $Y$ in the category $\operatorname{\mathcal{C}}_ n$ is an $n$-simplex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$. In other words, we have an equality of sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}_ n}(X,Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$.

  • For every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}_ n$, the composition $g \circ f: X \rightarrow Z$ is given by the image of the vertex $(g,f)$ under the map of simplicial sets

    \[ c_{Z,Y,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }. \]
  • For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}}_ n) = \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the identity morphism from $X$ to itself in the category $\operatorname{\mathcal{C}}_ n$ is the $n$-simplex of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet }$ which corresponds to the composite map

    \[ \Delta ^ n \rightarrow \Delta ^{0} \xrightarrow { \operatorname{id}_ X } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet } \]

Example 2.4.1.4 (The Underlying Category of a Simplicial Category). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We let $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ denote the ordinary category obtained by applying Construction 2.4.1.3 in the case $n=0$. We will refer to $\operatorname{\mathcal{C}}$ as the underlying category of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$. Note that $\operatorname{\mathcal{C}}$ can also be obtained from $\operatorname{\mathcal{C}}_{\bullet }$ by applying the general procedure described in Example 2.1.7.5.

We will sometimes abuse terminology by identifying a simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ with its underlying category $\operatorname{\mathcal{C}}$. In particular, if $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}_{\bullet }$, we will write $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to denote the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}_0}(X,Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{0}$ of morphisms from $X$ to $Y$ in the category $\operatorname{\mathcal{C}}$.

Example 2.4.1.5 (Topological Spaces). Let $\operatorname{Top}$ denote the category whose objects are topological spaces and whose morphisms are continuous functions. Then $\operatorname{Top}$ can be promoted to a simplicial category $\operatorname{Top}_{\bullet }$: given a pair of topological spaces $X$ and $Y$, we define the simplicial set $\operatorname{Hom}_{\operatorname{Top}}(X,Y)_{\bullet }$ informally by the formula

\[ \operatorname{Hom}_{\operatorname{Top}}(X,Y)_{n} = \operatorname{Hom}_{\operatorname{Top}}( | \Delta ^ n | \times X, Y) \]

In particular, a vertex of $\operatorname{Hom}_{\operatorname{Top}}(X,Y)_{\bullet }$ can be identified with a continuous function $f: X \rightarrow Y$. Moreover, for any topological space $Y$, we have a canonical isomorphism of simplicial sets $\operatorname{Hom}_{\operatorname{Top}}( \ast , Y )_{\bullet } \simeq \operatorname{Sing}_{\bullet }(Y)$, where $\operatorname{Sing}_{\bullet }(Y)$ is the singular simplicial set of Construction 1.1.7.1.

Let $\operatorname{\mathcal{C}}$ be a category. Roughly speaking, a simplicial enrichment $\operatorname{\mathcal{C}}_{\bullet }$ of $\operatorname{\mathcal{C}}$ can be viewed as a datum which allows us to “do homotopy theory” in $\operatorname{\mathcal{C}}$. For example, it allows us to define a notion of homotopy between morphisms of $\operatorname{\mathcal{C}}$:

Definition 2.4.1.6. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category, and let $f,g: X \rightarrow Y$ be two morphisms in the underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ having the same source and target. A homotopy from $f$ to $g$ is an edge $h \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{1}$ satisfying $d_1(h) = f$ and $d_0(h) = g$.

Example 2.4.1.7. Let $X$ and $Y$ be topological spaces and let $f,g: X \rightarrow Y$ be continuous functions, which we regard as morphisms in the simplicial category $\operatorname{Top}_{\bullet }$ of Example 2.4.1.5. Then a homotopy from $f$ to $g$ in the sense of Definition 2.4.1.6 is a homotopy in the usual sense: a continuous function $h: [0,1] \times X = | \Delta ^1 | \times X \rightarrow Y$ satisfying $h(0,x) = f(x)$ and $h(1,x) = g(x)$ for all $x \in X$.

In a general simplicial category $\operatorname{\mathcal{C}}$, the notion of homotopy (in the sense of Definition 2.4.1.6) need not be well-behaved: for example, the existence of a homotopy from $f$ to $g$ need not imply the existence of a homotopy from $g$ to $f$. To remedy the situation, it is convenient to restrict attention to a special class of simplicial categories:

Definition 2.4.1.8. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We will say that $\operatorname{\mathcal{C}}_{\bullet }$ is locally Kan if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}_{\bullet }$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is a Kan complex (Definition 1.1.9.1).

Remark 2.4.1.9. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a locally Kan simplicial category, and let $f,g: X \rightarrow Y$ be a pair of morphisms in the underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ having the same source and target. Invoking Proposition 1.1.9.10, we see that the following conditions are equivalent:

$(a)$

There exists a homotopy from $f$ to $g$, in the sense of Definition 2.4.1.6.

$(b)$

The morphisms $f$ and $g$ belong to the same connected component of the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.

In particular, condition $(a)$ defines an equivalence relation on the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.

Exercise 2.4.1.10. Let $\operatorname{Top}_{\bullet }$ be the simplicial category of Example 2.4.1.5. Show that $\operatorname{Top}_{\bullet }$ is locally Kan (hint: generalize the proof of Proposition 1.1.9.8).

Specializing Definition 2.1.7.10 to the setting of simplicial enrichments, we obtain the following:

Definition 2.4.1.11 (Simplicial Functors). Let $\operatorname{\mathcal{C}}_{\bullet }$ and $\operatorname{\mathcal{D}}_{\bullet }$ be simplicial categories. A simplicial functor $F: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \operatorname{\mathcal{D}}_{\bullet }$ consists of the following data:

$(1)$

For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, and object $F(X) \in \operatorname{Ob}(\operatorname{\mathcal{D}}_{\bullet })$.

$(2)$

For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, a map of simplicial sets $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )_{\bullet }$.

These data are required to satisfy the following conditions:

  • For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the map of simplicial sets $F_{X,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(X) )_{\bullet }$ carries the vertex $\operatorname{id}_ X$ to the the vertex $\operatorname{id}_{F(X)}$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the diagram of simplicial sets

    \[ \xymatrix { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [r] \ar [d]^{ F_{Y,Z} \otimes F_{X,Y} } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \ar [d]^{ F_{X,Z} } \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(Y), F(Z) )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )_{\bullet } \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Z) )_{\bullet } } \]

    is commutative.

We let $\operatorname{Cat_{\Delta }}$ denote the category whose objects are (small) simplicial categories and whose morphisms are simplicial functors.

Remark 2.4.1.12. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a (small) simplicial category. Then the construction $[n] \mapsto \operatorname{\mathcal{C}}_{n}$ determines a functor from the simplex category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ (Definition 1.1.1.2) to the category $\operatorname{Cat}$ of (small) categories. Allowing $\operatorname{\mathcal{C}}_{\bullet }$ to vary, we obtain a functor $\operatorname{Cat_{\Delta }}\rightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$, which fits into a pullback diagram of categories

\[ \xymatrix@C =80pt@R=50pt{ \operatorname{Cat_{\Delta }}\ar [r]^-{ \operatorname{\mathcal{C}}_{\bullet } \mapsto ([n] \mapsto \operatorname{\mathcal{C}}_ n)} \ar [d]^{\operatorname{Ob}} & \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat}) \ar [d]^{ \operatorname{Ob}} \\ \operatorname{Set}\ar [r] & \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set}), } \]

where the lower horizontal map carries each set $S$ to the constant functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ taking the value $S$.

Phrased more informally: simplicial categories can be identified with simplicial objects $\operatorname{\mathcal{C}}_{\bullet }$ of $\operatorname{Cat}$ for which the underlying simplicial set of objects $[n] \mapsto \operatorname{Ob}( \operatorname{\mathcal{C}}_{n} )$ is constant. In particular, the functor $\operatorname{Cat_{\Delta }}\rightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$ is a fully faithful embedding.

Proof. The category $\operatorname{Cat}$ admits small limits and colimits, which are preserved by the forgetful functor $\operatorname{Ob}: \operatorname{Cat}\rightarrow \operatorname{Set}$. It follows that the category $\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$ of simplicial objects in $\operatorname{Cat}$ also admits small limits and colimits, which are computed pointwise. Remark 2.4.1.12 supplies a fully faithful embedding $\operatorname{Cat_{\Delta }}\hookrightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$ whose essential image is closed under small limits and colimits, so that $\operatorname{Cat_{\Delta }}$ admits small limits and colimits as well. $\square$