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2.1.1 Nonunital Monoidal Categories

Let $\operatorname{Cat}$ denote the category whose objects are (small) categories and whose morphisms are functors. Then $\operatorname{Cat}$ admits finite products. One can therefore consider (nonunital) monoids in $\operatorname{Cat}$: that is, small categories $\operatorname{\mathcal{C}}$ equipped with a strictly associative multiplication $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$. For the convenience of the reader, we spell out this definition in detail (and abandon the smallness assumption on $\operatorname{\mathcal{C}}$):

Definition 2.1.1.1. Let $\operatorname{\mathcal{C}}$ be a category. A nonunital strict monoidal structure on $\operatorname{\mathcal{C}}$ is a functor

\[ \otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad (X,Y) \mapsto X \otimes Y \]

which is strictly associative in the following sense:

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, we have an equality $X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z$ (as objects of $\operatorname{\mathcal{C}}$).

  • For every triple of morphisms $f: X \rightarrow X'$, $g: Y \rightarrow Y'$, $h: Z \rightarrow Z'$, we have an equality

    \[ f \otimes (g \otimes h) = (f \otimes g) \otimes h \]

    of morphisms in $\operatorname{\mathcal{C}}$ from the object $X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z$ to the object $X' \otimes (Y' \otimes Z') = (X' \otimes Y') \otimes Z'$).

A nonunital strict monoidal category is a pair $(\operatorname{\mathcal{C}}, \otimes )$, where $\operatorname{\mathcal{C}}$ is a category and $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a nonunital strict monoidal structure on $\operatorname{\mathcal{C}}$.

Remark 2.1.1.2. We will often abuse terminology by identifying a nonunital strict monoidal category $(\operatorname{\mathcal{C}}, \otimes )$ with the underlying category $\operatorname{\mathcal{C}}$. If we refer to a category $\operatorname{\mathcal{C}}$ as a nonunital strict monoidal category, we implicitly assume that $\operatorname{\mathcal{C}}$ is has been endowed with a tensor product functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ which is strictly associative in the sense of Definition 2.1.1.1.

Example 2.1.1.3. Let $M$ be a set, which we regard as a category having only identity morphisms. Then nonunital strict monoidal structures on $M$ (in the sense of Definitions 2.1.1.1) can be identified with nonunital monoid structures on $M$ (in the sense of Definition 2.1.0.3). In particular, any nonunital monoid can be regarded as a strict nonunital monoidal category (having only identity morphisms).

Example 2.1.1.4 (Endomorphism Categories). Let $\operatorname{\mathcal{C}}$ be a category, and let $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ denote the category of functors from $\operatorname{\mathcal{C}}$ to itself. Then the composition functor

\[ \circ : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}) \times \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}) \mapsto \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}) \quad \quad (F,G) \mapsto F \circ G; \]

is a strict nonunital monoidal structure on $\operatorname{\mathcal{C}}$.

For many purposes, Definition 2.1.1.1 is too restrictive. Note that if $k$ is a field, then the tensor product functor $\otimes _{k}: \operatorname{Vect}_{k} \otimes \operatorname{Vect}_{k} \rightarrow \operatorname{Vect}_{k}$ of Example 2.1.0.2 does not quite fit the framework described in Definition 2.1.1.1. Given vector spaces $X$, $Y$, and $Z$ over $k$, there is no reason to expect the iterated tensor products $X \otimes _{k} (Y \otimes _{k} Z)$ and $(X \otimes _{k} Y) \otimes _{k} Z$ to be identical. In fact, this is impossible to determine based from the definition sketched in Example 2.1.0.2. To construct the functor $\otimes _{k}$ explicitly, we need to make certain choices: namely, a choice of universal bilinear map $b: U \times V \rightarrow U \otimes _{k} V$ for every pair of vector spaces $U,V \in \operatorname{Vect}_{k}$. Without an explicit convention for how these choices are to be made, we cannot answer the question of whether the vector spaces $X \otimes _{k} (Y \otimes _{k} Z)$ and $(X \otimes _{k} Y) \otimes _{k} Z$ are equal. However, this is arguably the wrong question to consider: in the setting of vector spaces, the appropriate notion of ÒsamenessÓ is not equality, but isomorphism. The iterated tensor products $X \otimes _{k} (Y \otimes _{k} Z)$ and $(X \otimes _{k} Y) \otimes _{k} Z$ are isomorphic, because they can be characterized by the same universal property: both are universal among vector spaces $W$ equipped with a $k$-trilinear map $t: X \times Y \times Z \rightarrow W$. Even better, there is a canonical isomorphism

\[ \alpha _{X,Y,Z}: X \otimes _{k} (Y \otimes _{k} Z) \rightarrow (X \otimes _{k} Y) \otimes _{k} Z, \]

which depends functorially on $X$, $Y$, and $Z$. Motivated by this example, we introduce the following generalization of Definition 2.1.1.1:

Definition 2.1.1.5. Let $\operatorname{\mathcal{C}}$ be a category. A nonunital monoidal structure on $\operatorname{\mathcal{C}}$ consists of the following data:

  • A functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$, which we will refer to as the tensor product functor.

  • A collection of isomorphisms $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$, for $X,Y,Z \in \operatorname{\mathcal{C}}$, called the associativity constraints of $\operatorname{\mathcal{C}}$. We demand that the associativity constraints $\alpha _{X,Y,Z}$ depend functorially on $X,Y,Z$ in the following sense: for every triple of morphisms $f: X \rightarrow X'$, $g: Y \rightarrow Y'$, and $h: Z \rightarrow Z'$, the diagram

    \[ \xymatrix@C =50pt@R=50pt{ X \otimes (Y \otimes Z) \ar [r]^-{ \alpha _{X,Y,Z } }_{\sim } \ar [d]^{ f \otimes (g \otimes h) } & (X \otimes Y) \otimes Z \ar [d]^{ (f \otimes g) \otimes h} \\ X' \otimes (Y' \otimes Z' ) \ar [r]^-{ \alpha _{X',Y',Z'} }_{\sim } & (X' \otimes Y') \otimes Z' } \]

    is commutative. In other words, we require that $\alpha = \{ \alpha _{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}}$ can be regarded as a natural isomorphism from the functor

    \[ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\xrightarrow { (X,Y,Z) \mapsto X \otimes (Y \otimes Z) } \operatorname{\mathcal{C}} \]

    to the functor

    \[ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\xrightarrow { (X,Y,Z) \mapsto (X \otimes Y) \otimes Z } \operatorname{\mathcal{C}}. \]

The associativity constraints of $\operatorname{\mathcal{C}}$ are required to satisfy the following additional condition:

$(P)$

For every quadruple of objects $W,X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram of isomophisms

\[ \xymatrix@C =-30pt@R=30pt{ & W \otimes ((X \otimes Y) \otimes Z) \ar [rr]^-{ \alpha _{W, X \otimes Y, Z} }_{\sim } & & (W \otimes (X \otimes Y)) \otimes Z \ar [dr]^{ \alpha _{W, X, Y} \otimes \operatorname{id}_ Z}_{\sim } & \\ W \otimes (X \otimes (Y \otimes Z) ) \ar [ur]^{ \operatorname{id}_ W \otimes \alpha _{X,Y,Z} }_{\sim } \ar [drr]_{ \alpha _{W,X,Y \otimes Z}}^{\sim } & & & & ((W \otimes X) \otimes Y) \otimes Z \\ & & (W \otimes X) \otimes (Y \otimes Z) \ar [urr]_{\alpha _{ W \otimes X, Y, Z} }^{\sim } & & } \]

commutes.

A nonunital monoidal category is a triple $(\operatorname{\mathcal{C}}, \otimes , \alpha )$, where $\operatorname{\mathcal{C}}$ is a category and $(\otimes , \alpha )$ is a nonunital monoidal structure on $\operatorname{\mathcal{C}}$.

Remark 2.1.1.6. In the setting of Definition 2.1.1.5, we will refer to $(P)$ as the pentagon identity. It is a prototypical example of a coherence condition: the existence of the associativity constraints $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$ guarantees that the tensor product is associative up to isomorphism, and the pentagon identity is a sort of “higher order” associative law satisfied by the isomorphisms themselves.

Example 2.1.1.7. Let $\operatorname{\mathcal{C}}$ be a category equipped with a nonunital strict monoidal structure $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 2.1.1.1). Then $\otimes $ determines a nonunital monoidal structure on $\operatorname{\mathcal{C}}$ (in the sense of Definition 2.1.1.5) by taking the associativity constraints $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$ to be identity morphisms. Conversely, if $\operatorname{\mathcal{C}}$ is equipped with a nonunital monoidal structure $(\otimes , \alpha )$ where each of the associativity constraints $\alpha _{X,Y,Z}$ is an identity morphism, then $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a nonunital strict monoidal structure on $\operatorname{\mathcal{C}}$.

Remark 2.1.1.8. Let $\operatorname{\mathcal{C}}$ be a category equipped with a nonunital monoidal structure $(\otimes , \alpha )$. We will often abuse terminology by identifying the nonunital monoidal structure $(\otimes , \alpha )$ with the underlying tensor product functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$. If we refer to a functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}$ as a nonunital monoidal structure on $\operatorname{\mathcal{C}}$, we implicitly assume that $\operatorname{\mathcal{C}}$ has been equipped with associativity constraints $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$ satisfying the pentagon identity of Definition 2.1.1.5. Beware that, in the non-strict case, the associativity constraints are an essential part of the data: it is possible to have inequivalent nonunital monoidal categories $(\operatorname{\mathcal{C}}, \otimes , \alpha )$ and $(\operatorname{\mathcal{C}}', \otimes ', \alpha ')$ with $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}'$ and $\otimes = \otimes '$ (see Example 2.1.3.3).