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2.1.2 Monoidal Categories

We now introduce unital versions of Definitions 2.1.1.1 and 2.1.1.5.

Definition 2.1.2.1. Let $\operatorname{\mathcal{C}}$ be a category. A strict monoidal structure on $\operatorname{\mathcal{C}}$ is a nonunital strict monoidal structure $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ for which there exists an object $\mathbf{1} \in \operatorname{\mathcal{C}}$ satisfying the following condition:

$(\ast )$

For every object $X \in \operatorname{\mathcal{C}}$, we have $X \otimes \mathbf{1} = X = \mathbf{1} \otimes X$ (as objects of $\operatorname{\mathcal{C}}$). Moreover, for every morphism $f: X \rightarrow X'$ in $\operatorname{\mathcal{C}}$, we have $f \otimes \operatorname{id}_{\bf {1}} = f = \operatorname{id}_{ \bf {1} } \otimes f$ (as morphisms from $X$ to $X'$).

A 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 strict monoidal structure on $\operatorname{\mathcal{C}}$.

Remark 2.1.2.2. Let $\operatorname{\mathcal{C}}$ be a nonunital strict monoidal category. We will say that an object $\mathbf{1} \in \operatorname{\mathcal{C}}$ is a strict unit if it satisfies condition $(\ast )$ of Definition 2.1.2.1. Note that if such an object exists, then it is uniquely determined: it can be characterized as the unit element of the monoid $\operatorname{Ob}(\operatorname{\mathcal{C}})$.

It follows from Remark 2.1.2.2 that the notion of strict unit is not invariant under isomorphism. To address this, it will be convenient to consider a more general notion of unit object, which makes sense in the non-strict setting as well. We will use an efficient formulation due to Saavedra ([MR0338002]); see also [MR2388233]. To motivate the definition, we begin with a simple observation about units in a more elementary setting.

Proposition 2.1.2.3. Let $M$ be a nonunital monoid, let $e$ be an element of $M$, and let $\ell _{e}: M \rightarrow M$ denote the function given by the formula $\ell _{e}(x) = ex$. The following conditions are equivalent:

$(a)$

The element $e$ is a left unit of $M$: that is, $\ell _{e}$ is the identity function from $M$ to itself.

$(b)$

The element $e$ is idempotent (that is, it satisfies $ee = e$) and the function $\ell _{e}: M \rightarrow M$ is a bijection.

$(c)$

The element $e$ is idempotent and the function $\ell _{e}: M \rightarrow M$ is a monomorphism.

Proof. The implications $(a) \Rightarrow (b) \Rightarrow (c)$ are immediate. To complete the proof, assume that $e$ satisfies condition $(c)$ and let $x$ be an element of $M$. Using the assumption that $e$ is idempotent (and the associativity of the multiplication on $M$), we obtain an identity $\ell _{e}(x) = ex = (ee)x = e(ex) = \ell _{e}( ex )$. Since $\ell _{e}$ is a monomorphism, it follows that $x = ex$. $\square$

Corollary 2.1.2.4. Let $M$ be a nonunital monoid. Then an element $e \in M$ is a unit if and only if the following conditions are satisfied:

$(i)$

The element $e$ is idempotent: that is, we have $ee = e$.

$(ii)$

The element $e$ is left cancellative: that is, the function $x \mapsto ex$ is a monomorphism from $M$ to itself.

$(iii)$

The element $e$ is right cancellative: that is, the function $x \mapsto xe$ is a monomorphism from $M$ to itself.

We now adapt the characterization of Corollary 2.1.2.4 to the setting of nonunital monoidal categories.

Definition 2.1.2.5. Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category. A unit of $\operatorname{\mathcal{C}}$ is a pair $( \mathbf{1}, \upsilon )$, where $\mathbf{1}$ is an object of $\operatorname{\mathcal{C}}$ and $\upsilon : \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$ is an isomorphism, which satisfies the following additional condition:

$(\ast )$

The functors

\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad C \mapsto \mathbf{1} \otimes C \]
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad C \mapsto C \otimes \mathbf{1} \]

are fully faithful.

Remark 2.1.2.6. Condition $(\ast )$ of Definition 2.1.2.5 depends only on the object $\mathbf{1} \in \operatorname{\mathcal{C}}$, and not on the choice of isomorphism $\upsilon : \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$.

Example 2.1.2.7. Let $\operatorname{\mathcal{C}}$ be a strict monoidal category, and let $\mathbf{1} \in \operatorname{\mathcal{C}}$ be the strict unit (Remark 2.1.2.2). Then $( \mathbf{1}, \operatorname{id}_{ \mathbf{1} } )$ is a unit of $\operatorname{\mathcal{C}}$.

Example 2.1.2.8. Let $M$ be a nonunital monoid, regarded as a (strict) nonunital monoidal category having only identity morphisms (Example 2.1.1.3). Then the converse of Example 2.1.2.7 holds: a pair $(\mathbf{1}, \upsilon )$ is a unit structure on $M$ (in the sense of Definition 2.1.2.5 if and only if $\mathbf{1}$ is a unit element of $M$ and $\upsilon = \operatorname{id}_{ \mathbf{1} }$. This is a restatement of Corollary 2.1.2.4.

If $M$ is a nonunital monoid, then a unit element $e \in M$ is unique if it exists. For nonunital monoidal categories, the analogous statement is more subtle. If a nonunital monoidal category $\operatorname{\mathcal{C}}$ admits a unit $(\mathbf{1}, \upsilon )$, then it has many others: we can replace $\mathbf{1}$ by any object $\mathbf{1}'$ which is isomorphic to it, and $\upsilon $ by any choice of isomorphism $\upsilon ': \mathbf{1}' \otimes \mathbf{1}' \simeq \mathbf{1}'$. Nevertheless, we have the following strong uniqueness result:

Proposition 2.1.2.9 (Uniqueness of Units). Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category equipped with units $( \mathbf{1}, \upsilon )$ and $( \mathbf{1}', \upsilon ')$ (in the sense of Definition 2.1.2.5). Then there is a unique isomorphism $u: \mathbf{1} \xrightarrow {\sim } \mathbf{1}'$ for which the diagram

\[ \xymatrix { \mathbf{1} \otimes \mathbf{1} \ar [d]^-{u \otimes u} \ar [r]^-{ \upsilon } & \mathbf{1} \ar [d]^{u} \\ \mathbf{1'} \otimes \mathbf{1'} \ar [r]^-{ \upsilon '} & \mathbf{1'} } \]

commutes.

We will give the proof of Proposition 2.1.2.9 at the end of this section.

Definition 2.1.2.10. Let $\operatorname{\mathcal{C}}$ be a category. A monoidal structure on $\operatorname{\mathcal{C}}$ is a nonunital monoidal structure $(\otimes , \alpha )$ on $\operatorname{\mathcal{C}}$ (Definition 2.1.1.5) together with a choice of unit $(\mathbf{1}, \upsilon )$ (in the sense of Definition 2.1.2.5). A monoidal category is a category $\operatorname{\mathcal{C}}$ together with a monoidal structure $( \otimes , \alpha , \mathbf{1}, \upsilon )$ on $\operatorname{\mathcal{C}}$. In this case, we refer to $\mathbf{1}$ as the unit object of $\operatorname{\mathcal{C}}$ and the isomorphism $\upsilon : \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$ as the unit constraint of $\operatorname{\mathcal{C}}$.

Remark 2.1.2.11. It is possible to adopt the following variant of Definition 2.1.2.10:

  • A monoidal category is a nonunital monoidal category $\operatorname{\mathcal{C}}$ which admits a unit, in the sense of Definition 2.1.2.5.

This is essentially equivalent to Definition 2.1.2.10, since a unit $(\mathbf{1}, \upsilon )$ of $\operatorname{\mathcal{C}}$ is uniquely determined up to unique isomorphism (Proposition 2.1.2.9). However, for our purposes it will be more convenient to adopt the convention that a monoidal structure on a category $\operatorname{\mathcal{C}}$ includes a choice of unit object $\mathbf{1} \in \operatorname{\mathcal{C}}$ and unit constraint $\upsilon : \mathbf{1} \otimes \mathbf{1} \simeq \mathbf{1}$.

Remark 2.1.2.12. Let $\operatorname{\mathcal{C}}$ be a category. We will sometimes abuse terminology by identifying a monoidal structure $(\otimes , \alpha , \mathbf{1}, \upsilon )$ with the underlying nonunital monoidal structure $(\otimes , \alpha )$ on $\operatorname{\mathcal{C}}$ (or with the underlying tensor product functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$). This is essentially harmless, by virtue of Remark 2.1.2.11. We will also abuse terminology (in a less harmless way) by identifying a monoidal category $(\operatorname{\mathcal{C}}, \otimes , \alpha , \mathbf{1}, \upsilon )$ with the underlying category $\operatorname{\mathcal{C}}$.

Notation 2.1.2.13. Let $\operatorname{\mathcal{C}}$ be a monoidal category. We will generally use the symbol $\mathbf{1}$ to denote the unit object of $\operatorname{\mathcal{C}}$. In situations where this notation is potentially confusing (for example, if we are comparing $\operatorname{\mathcal{C}}$ with another monoidal category), we will often disambiguate by instead writing $\mathbf{1}_{\operatorname{\mathcal{C}}}$ for the unit object of $\operatorname{\mathcal{C}}$.

Example 2.1.2.14. Let $\operatorname{\mathcal{C}}$ be a category. Then every strict monoidal structure $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 2.1.2.1) can be promoted to a monoidal structure $(\otimes , \alpha , \mathbf{1}, \upsilon )$ on $\operatorname{\mathcal{C}}$, by taking $\mathbf{1}$ to be the strict unit of $\operatorname{\mathcal{C}}$ and the associativity and unit constraints to be identity morphisms of $\operatorname{\mathcal{C}}$. Conversely, if $\operatorname{\mathcal{C}}$ is equipped with a monoidal structure $(\otimes , \alpha , \mathbf{1}, \upsilon )$ for which the associativity and unit constraints are identity morphisms, then $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a strict monoidal structure on $\operatorname{\mathcal{C}}$ and $\mathbf{1}$ is the strict unit.

Let $\operatorname{\mathcal{C}}$ be a monoidal category. In general, the unit object $\mathbf{1}$ of $\operatorname{\mathcal{C}}$ need not be strict, in the sense that the functors

\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad X \mapsto \mathbf{1} \otimes X \]
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad X \mapsto X \otimes \mathbf{1} \]

need not be equal to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$. However, they are always (canonically) isomorphic to $\operatorname{id}_{\operatorname{\mathcal{C}}}$.

Construction 2.1.2.15 (Left and Right Unit Constraints). Let $\operatorname{\mathcal{C}}= (\operatorname{\mathcal{C}}, \otimes , \alpha , \mathbf{1}, \upsilon )$ be a monoidal category. For each object $X \in \operatorname{\mathcal{C}}$, we have canonical isomorphisms

\[ \mathbf{1} \otimes ( \mathbf{1} \otimes X) \xrightarrow { \alpha _{ \mathbf{1}, \mathbf{1}, X} } ( \mathbf{1} \otimes \mathbf{1} ) \otimes X \xrightarrow { \upsilon \otimes \operatorname{id}_ X} \mathbf{1} \otimes X. \]

Since the functor $Y \mapsto \mathbf{1} \otimes Y$ is fully faithful, it follows that that there is a unique isomorphism $\lambda _{X}: \mathbf{1} \otimes X \xrightarrow {\sim } X$ for which the diagram

\[ \xymatrix { \mathbf{1} \otimes (\mathbf{1} \otimes X) \ar [rr]^-{\alpha _{\mathbf{1}, \mathbf{1}, X} }_{\sim } \ar [dr]_{ \operatorname{id}_\mathbf {1} \otimes \lambda _ X}^{\sim } & & (\mathbf{1} \otimes \mathbf{1}) \otimes X \ar [dl]^{ \upsilon \otimes \operatorname{id}_ X}_{\sim } \\ & \mathbf{1} \otimes X & } \]

commutes. We will refer to $\lambda _{X}$ as the left unit constraint. Similarly, there is a unique isomorphism $\rho _{X}: X \otimes \mathbf{1} \simeq X$ for which the diagram

\[ \xymatrix { X \otimes (\mathbf{1} \otimes \mathbf{1}) \ar [rr]^-{\alpha _{X, \mathbf{1}, \mathbf{1}} }_{\sim } \ar [dr]_{ \operatorname{id}_ X \otimes \upsilon }^{\sim } & & (X \otimes \mathbf{1}) \otimes \mathbf{1} \ar [dl]^{ \rho _ X \otimes \operatorname{id}_{ \mathbf{1} } }_{\sim } \\ & X \otimes \mathbf{1} & } \]

commutes; we refer to $\rho _ X$ as the right unit constraint.

Remark 2.1.2.16. Let $\operatorname{\mathcal{C}}$ be a monoidal category. Then the left and right unit constraints $\lambda _{X}: \mathbf{1} \otimes X \xrightarrow {\sim } X$ and $\rho _{X}: X \otimes \mathbf{1} \xrightarrow {\sim } X$ depend functorially on $X$. In other words, for every morphism $f: X \rightarrow Y$, the diagram

\[ \xymatrix@R =50pt@C=50pt{ \mathbf{1} \otimes X \ar [d]^-{ \operatorname{id}_{ \mathbf{1}} \otimes f } \ar [r]^-{ \lambda _{X} }_{\sim } & X \ar [d]^{ f } & X \otimes \mathbf{1} \ar [l]_-{ \rho _ X}^{\sim } \ar [d]^{f \otimes \operatorname{id}_{ \mathbf{1} }}\\ \mathbf{1} \otimes Y \ar [r]^-{ \lambda _{Y} }_{\sim } & Y & Y \otimes \mathbf{1} \ar [l]_-{ \rho _ Y}^{\sim } } \]

is commutative.

Proposition 2.1.2.17 (The Triangle Identity). Let $\operatorname{\mathcal{C}}$ be a monoidal category with unit object $\mathbf{1}$. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $\rho _{X}: X \otimes \mathbf{1} \simeq X$ and $\lambda _{Y}: \mathbf{1} \otimes Y \rightarrow Y$ be the right and left unit constraints of Construction 2.1.2.15. Then the diagram of isomorphisms

\[ \xymatrix { X \otimes (\mathbf{1} \otimes Y) \ar [rr]^-{\alpha _{X, \mathbf{1}, Y} }_{\sim } \ar [dr]_{ \operatorname{id}_ X \otimes \lambda _ Y}^{\sim } & & (X \otimes \mathbf{1}) \otimes Y \ar [dl]^{ \rho _{X} \otimes \operatorname{id}_ Y}_{\sim } \\ & X \otimes Y & } \]

is commutative.

Proof. We have a diagram of isomorphisms

\[ \xymatrix@R =50pt@C=-15pt{ & X \otimes ( (\mathbf{1} \otimes \mathbf{1}) \otimes Y) ) \ar [rr]^{\alpha } \ar [d]^{\upsilon } & & (X \otimes ( \mathbf{1} \otimes \mathbf{1} ) ) \otimes Y \ar [d]^{\upsilon } \ar [ddr]^{\alpha } & \\ & X \otimes (\mathbf{1} \otimes Y) \ar [rr]^{ \alpha } \ar [d]^{\alpha } & & (X \otimes \mathbf{1}) \otimes Y \ar [dll]^{ \operatorname{id}} & \\ X \otimes (\mathbf{1} \otimes (\mathbf{1} \otimes Y) ) \ar [uur]^{\alpha } \ar [ur]_{\lambda _ Y} \ar [drr]^{ \alpha } & (X \otimes \mathbf{1}) \otimes Y \ar [r]^-{ \rho _{X}} & X \otimes Y & X \otimes ( \mathbf{1} \otimes Y) \ar [l]_-{ \lambda _ Y} \ar [u]_{ \alpha } & ((X \otimes \mathbf{1}) \otimes \mathbf{1}) \otimes Y \ar [ul]_{\rho _ X} \\ & & (X \otimes \mathbf{1}) \otimes (\mathbf{1} \otimes Y). \ar [ul]_{ \lambda _ Y } \ar [ur]^{ \rho _ X } \ar [urr]^{ \alpha } & & } \]

Here the outer cycle commutes by the pentagon identity $(P)$ of Definition 2.1.1.5, the upper rectangle by the functoriality of the associativity constraint, the upper side triangles by the definition of the left and right unit constraints, the quadrilaterals on the lower sides by Remark 2.1.2.16, and the lower region by the functoriality of the tensor product $\otimes $. It follows that the middle square is also commutative, which is equivalent to the statement of Proposition 2.1.2.17. $\square$

Exercise 2.1.2.18. Let $\operatorname{\mathcal{C}}$ be a monoidal category. with unit object $\mathbf{1}$. Show that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the diagrams

\[ \xymatrix { X \otimes (Y \otimes \mathbf{1}) \ar [rr]^{\alpha _{X,Y,\mathbf{1}} } \ar [dr]_{\operatorname{id}_ X \otimes \rho _{Y}} & & (X \otimes Y) \otimes \mathbf{1} \ar [dl]^{ \rho _{ X \otimes Y} } \\ & X \otimes Y & } \]
\[ \xymatrix { \mathbf{1} \otimes (X \otimes Y) \ar [dr]_{\lambda _{X \otimes Y}} \ar [rr]^{ \alpha _{ \mathbf{1}, X, Y} } & & ( \mathbf{1} \otimes X) \otimes Y \ar [dl]^{ \lambda _ X \otimes \operatorname{id}_ Y} \\ & X \otimes Y & } \]

are commutative (for a more general statement, see Proposition 2.2.1.16.

Corollary 2.1.2.19. Let $\operatorname{\mathcal{C}}$ be a monoidal category with unit object $\mathbf{1}$. Then the left and right unit constraints $\lambda _{ \mathbf{1} }, \rho _{ \mathbf{1} }: \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$ are equal to the unit constraint $\upsilon : \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$.

Proof. Let $X$ be any object of $\operatorname{\mathcal{C}}$. Then the left unit contraint $\lambda _{X}$ is characterized by the commutativity of the diagram

\[ \xymatrix { \mathbf{1} \otimes (\mathbf{1} \otimes X) \ar [rr]^-{\alpha _{\mathbf{1}, \mathbf{1}, X} }_{\sim } \ar [dr]_{ \operatorname{id}_\mathbf {1} \otimes \lambda _ X}^{\sim } & & (\mathbf{1} \otimes \mathbf{1}) \otimes X \ar [dl]^{ \upsilon \otimes \operatorname{id}_ X}_{\sim } \\ & \mathbf{1} \otimes X. & } \]

Using Proposition 2.1.2.17, we deduce that $\upsilon \otimes \operatorname{id}_{X} = \rho _{ \mathbf{1} } \otimes \operatorname{id}_ X$ as morphisms from $( \bf {1} \otimes \mathbf{1}) \otimes X$ to $\mathbf{1} \otimes X$. In other words, the morphisms $\upsilon , \rho _\mathbf {1}: \mathbf{1} \otimes \mathbf{1} \rightarrow \mathbf{1}$ have the same image under the functor

\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad Y \mapsto Y \otimes X. \]

In the case $X = \mathbf{1}$, this functor is fully faithful; it follows that $\upsilon = \rho _\mathbf {1}$. The equality $\upsilon = \lambda _{ \mathbf{1} }$ follows by a similar argument. $\square$

Proof of Proposition 2.1.2.9. Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category equipped with units $(\mathbf{1}, \upsilon )$ and $(\mathbf{1}', \upsilon ')$. We can then regard $\operatorname{\mathcal{C}}$ as a monoidal category with unit object $\mathbf{1}$ and unit constraint $\upsilon $. For each object $X \in \operatorname{\mathcal{C}}$, let $\lambda _{X}: \mathbf{1} \otimes X \xrightarrow {\sim } X$ be the left unit constraint of Construction 2.1.2.15. We wish to show that there is a unique isomorphism $u: \mathbf{1} \simeq \mathbf{1}'$ for which the outer rectangle in the diagram of isomorphisms

\[ \xymatrix@C =50pt@R=50pt{ \mathbf{1} \otimes \mathbf{1} \ar [r]^-{ \lambda _{ \mathbf{1} } } \ar [d]^{ \operatorname{id}_{ \mathbf{1} } \otimes u } & \mathbf{1} \ar [d]^{u} \\ \mathbf{1} \otimes \mathbf{1}' \ar [r]^{ \lambda _{ \mathbf{1}' }} \ar [d]^{u \otimes \operatorname{id}_{ \mathbf{1}' }} & \mathbf{1}' \ar [d]^{\operatorname{id}_{ \mathbf{1}' }} \\ \mathbf{1}' \otimes \mathbf{1}' \ar [r]^{ \upsilon ' } & \mathbf{1}' } \]

is commutative. Since the upper square commutes (Remark 2.1.2.16), this is equivalent to the commutativity of the lower square. The existence and uniqueness of $u$ now follows from the assumption that the functor $X \mapsto X \otimes \mathbf{1}'$ is fully faithful. $\square$

Remark 2.1.2.20. Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category. Suppose we are given objects $\mathbf{1}, \mathbf{1}' \in \operatorname{\mathcal{C}}$ together with isomorphisms

\[ \upsilon : \mathbf{1} \otimes \mathbf{1} \simeq \mathbf{1} \quad \quad \upsilon ': \mathbf{1}' \otimes \mathbf{1}' \simeq \mathbf{1}'. \]

To carry out the proof of Proposition 2.1.2.9, it is sufficient to assume that the functors

\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad X \mapsto \mathbf{1} \otimes X \]
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad X \mapsto X \otimes \mathbf{1}' \]

are fully faithful: the first assumption is sufficient to construct the left unit constraints of Construction 2.1.2.15, and the second is used at the end of the proof. This can be regarded as a categorical analogue of the observation that if a nonunital monoid admits a left unit $e$ and a right unit $e'$, then we must have $e = e'$.