2.1 Monoidal Categories
Recall that a monoid is a set $M$ equipped with a map
\[ m: M \times M \rightarrow M \quad \quad (x,y) \mapsto xy \]
which satisfies the following conditions:
 $(a)$
The multiplication $m$ is associative. That is, we have $x(yz) = (xy)z$ for each triple of elements $x,y,z \in M$.
 $(b)$
There exists an element $e \in M$ such that $ex=x=xe$ for each $x \in M$ (in this case, the element $e$ is uniquely determined; we refer to it as the unit element of $M$).
Monoids are ubiquitous in mathematics:
Example 2.1.0.1. Let $\operatorname{\mathcal{C}}$ be a category and let $X$ be an object of $\operatorname{\mathcal{C}}$. An endomorphism of $X$ is a morphism from $X$ to itself in the category $\operatorname{\mathcal{C}}$. We let $\operatorname{End}_{\operatorname{\mathcal{C}}}(X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$ denote the set of all endomorphisms of $X$. The composition law on $\operatorname{\mathcal{C}}$ determines a map
\[ \operatorname{End}_{\operatorname{\mathcal{C}}}(X) \times \operatorname{End}_{\operatorname{\mathcal{C}}}(X) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}}(X) \quad \quad (f,g) \mapsto f \circ g, \]
which exhibits $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ as a monoid; the unit element of $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ is the identity morphism $\operatorname{id}_{X}: X \rightarrow X$. We refer to $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ as the endomorphism monoid of $X$.
In the setting of category theory, one often encounters multiplication laws which satisfy a more subtle form of associativity.
Example 2.1.0.2. Let $k$ be a field and let $U$, $V$, and $W$ be vector spaces over $k$. Recall that a function $b: U \times V \rightarrow W$ is said to be $k$bilinear if it satisfies the identities
\[ b(u+u',v) = b(u,v) + b(u',v) \quad \quad b(u,v + v') = b(u,v) + b(u,v') \]
\[ b( \lambda u, v) = \lambda b(u, v) = b(u, \lambda v) \text{ for $\lambda \in k$.} \]
We say that a $k$bilinear map $b: U \times V \rightarrow W$ is universal if, for any $k$vector space $W'$, composition with $b$ induces a bijection
\[ \{ \text{$k$linear maps $W \rightarrow W'$} \} \simeq \{ \text{$k$bilinear maps $U \times V \rightarrow W'$} \} . \]
If this condition is satisfied, then $W$ is determined (up to unique isomorphism) by $U$ and $V$; we refer to $W$ as the tensor product of $U$ and $V$ and denote it by $U \otimes _{k} V$ . The construction $(U,V) \mapsto U \otimes _{k} V$ then determines a functor
\[ \otimes _{k}: \operatorname{Vect}_{k} \times \operatorname{Vect}_{k} \rightarrow \operatorname{Vect}_{k}, \]
which we will refer to as the tensor product functor. It is associative in the following sense: for every triple of vector spaces $U,V,W \in \operatorname{Vect}_{k}$, there exists a canonical isomorphism
\[ U \otimes _{k} (V \otimes _{k} W) \xrightarrow {\sim } (U \otimes _{k} V) \otimes _{k} W \quad \quad u \otimes (v \otimes w) \mapsto (u \otimes v) \otimes w. \]
Our goal in this section is to review the theory of monoidal categories, which axiomatizes the essential features of Example 2.1.0.2. To simplify the discussion, we begin by developing the nonunital version of this theory.
Definition 2.1.0.3. A nonunital monoid is a set $M$ equipped with a map
\[ m: M \times M \rightarrow M \quad \quad (x,y) \mapsto xy \]
which satisfies the associative law $x(yz) = (xy)z$ for $x,y,z \in M$.
Warning 2.1.0.4. The terminology of Definition 2.1.0.3 is not standard. Most authors use the term semigroup for what we call a nonunital monoid.
In §2.1.1, we generalize Definition 2.1.0.3 by introducing the notion of a nonunital monoidal structure on a category $\operatorname{\mathcal{C}}$ (Definition 2.1.1.5). Roughly speaking, a nonunital monoidal structure on $\operatorname{\mathcal{C}}$ is a tensor product functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ which is associative up to isomorphism. More precisely, it consists of the functor $\otimes $ together with a choice of isomorphism $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \xrightarrow {\sim } (X \otimes Y) \otimes Z$ for every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$ (these isomorphisms are called the associativity constraints of $\operatorname{\mathcal{C}}$). The isomorphisms $\alpha _{X,Y,Z}$ are required to depend functorially on $X$, $Y$, and $Z$, and to satisfy a further coherence condition called the pentagon identity (this condition was introduced by MacLane in [MR0170925], and is sometimes known as MacLane's pentagon identity).
By definition, a nonunital monoid $M$ is a monoid if and only if there exists an element $e \in M$ satisfying $ex = x = xe$ for each $x \in M$. If this condition is satisfied, then the element $e$ is uniquely determined. The categorical analogue of this statement is a bit more subtle. Let $X$ be an object of a nonunital monoidal category $\operatorname{\mathcal{C}}$, and let $\ell _{X}, r_{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ denote the functors given by $\ell _{X}(Y) = X \otimes Y$ and $r_{X}(Y) = Y \otimes X$. In §2.1.2, we define a unit in $\operatorname{\mathcal{C}}$ to be an object $\mathbf{1}$ with the property that the functors $\ell _{ \mathbf{1} }$ and $r_{ \mathbf{1}}$ are fully faithful, together with a choice of isomorphism $\upsilon : \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$. In this case, the pair $(\mathbf{1}, \upsilon )$ is not unique; however, it is unique up to (unique) isomorphism (Proposition 2.1.2.9). One can use $\upsilon $ to construct natural isomorphisms
\[ \lambda _{Y}: \mathbf{1} \otimes Y \xrightarrow {\sim } Y \quad \quad \rho _{Y}: Y \otimes \mathbf{1} \xrightarrow {\sim } Y, \]
so that $\mathbf{1}$ really behaves like a unit for the tensor product $\otimes $ (Construction 2.1.2.17). We define a monoidal category to be a nonunital monoidal category $\operatorname{\mathcal{C}}$ together with a choice of unit $( \mathbf{1}, \upsilon )$ (Definition 2.1.2.10). A basic prototype is the category $\operatorname{Vect}_{k}$ of vector spaces over a field $k$ (equipped with the tensor product and associativity constraints given in Example 2.1.0.2, and the unit given by the object $k \in \operatorname{Vect}_{k}$). We give a more detailed description of this and other examples in §2.1.3.
The collection of (nonunital) monoids can be organized into a category:
Definition 2.1.0.5. Let $M$ and $M'$ be nonunital monoids. We say that a function $f: M \rightarrow M'$ is a nonunital monoid homomorphism if, for every pair of elements $x,y \in M$, we have $f(xy) = f(x) f(y)$. If $M$ and $M'$ are monoids, we say that $f$ is a monoid homomorphism if it is a nonunital monoid homomorphism which carries the unit element $e \in M$ to the unit element $e' \in M'$.
We let $\operatorname{Mon}^{\operatorname{nu}}$ denote the category whose objects are nonunital monoids and whose morphisms are nonunital monoid homomorphisms, and $\operatorname{Mon}\subset \operatorname{Mon}^{\operatorname{nu}}$ the subcategory whose objects are monoids and whose morphisms are monoid homomorphisms.
Most of the rest of this section is devoted to studying categorytheoretic analogues of Definition 2.1.0.5. We start in §2.1.4 with the nonunital case. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$ are nonunital monoidal categories, we define a nonunital monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{C}}'$ to be a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ together with a collection of isomorphisms
\[ \mu _{X,Y}: F(X) \otimes F(Y) \xrightarrow {\sim } F(X \otimes Y), \]
which depend functorially on $X,Y \in \operatorname{\mathcal{C}}$ and are compatible with the associativity constraints on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$ (Definition 2.1.4.4). We also introduce the more general notion of nonunital lax monoidal functor, where we do not require the morphisms $\mu _{X,Y}$ to be isomorphisms (Definition 2.1.4.3). Both of these definitions have unital analogues, which we study in §2.1.6 and §2.1.5, respectively.
We conclude this section in §2.1.7 with a brief review of enriched category theory. If $\operatorname{\mathcal{A}}$ is a monoidal category, then an $\operatorname{\mathcal{A}}$enriched category $\operatorname{\mathcal{C}}$ consists of a collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$ of objects of $\operatorname{\mathcal{C}}$, a collection of mapping objects $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \in \operatorname{\mathcal{A}}$ for each pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, and a composition law
\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \]
which is required to be unital and associative (see Definition 2.1.7.1). Enriched category theory will play an important role throughout this chapter: we will be particularly interested in the special case where $\operatorname{\mathcal{A}}= \operatorname{Cat}$ is the category of small categories (in which case we recover the notion of strict $2$category, which we study in §2.2), where $\operatorname{\mathcal{A}}= \operatorname{Set_{\Delta }}$ is the category of simplicial sets (in which case we recover the notion of simplicial category, which we study in §2.4), and where $\operatorname{\mathcal{A}}= \operatorname{Ch}(\operatorname{\mathbf{Z}})( \operatorname{ Ab })$ is the category of chain complexes of abelian groups (In which case we recover the notion of differential graded category, which we study in §2.5.
Structure

Subsection 2.1.1: Nonunital Monoidal Categories

Subsection 2.1.2: Monoidal Categories

Subsection 2.1.3: Examples of Monoidal Categories

Subsection 2.1.4: Nonunital Monoidal Functors

Subsection 2.1.5: Lax Monoidal Functors

Subsection 2.1.6: Monoidal Functors

Subsection 2.1.7: Enriched Category Theory