# nLab distributive monoidal category

Distributive monoidal categories

For the cartesian case see at distributive category.

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Distributive monoidal categories

## Idea

A distributive monoidal category is a monoidal category whose tensor product distributes over coproducts.

## Definition

A distributive monoidal category (this is not entirely standard terminology) is a monoidal category with coproducts whose tensor product preserves coproducts in each variable: i.e. such that the canonical morphisms

$\coprod_i (X\otimes Y_i)\to X \otimes \coprod_i Y_i$
$\coprod_i (X_i\otimes Y)\to \coprod_i X_i \otimes Y$

are isomorphisms.

Depending on the arity of the coproducts in question, we may speak of a finitary or infinitary distributive monoidal category.

The special case of distributive cartesian monoidal categories is known simply as distributive categories. Conversely, distributive monoidal categories are a special case of rig categories.

A more abstract way to say this, due to Weber and Batanin, is that if $M$ is the free monoidal category monad and $M V \xrightarrow{\otimes} V$ is the structure map of a monoidal category $V$, then $V$ is distributive if it admits left Kan extensions along functors $f:A\to B$ between discrete categories (of some size), and moreover if

$\array{ A && \xrightarrow{f} && B\\ & \searrow & \xRightarrow{\phi} & \swarrow\\ && V }$

is such a Kan extension, then so is

$\array{ M A && \xrightarrow{M f} && M B\\ & \searrow & \xRightarrow{M \phi} & \swarrow\\ && M V\\ && \downarrow^\otimes\\ && V. }$

## Examples

### Various

Beyond distributive categories, examples of distributive monoidal categories include the following:

In all these cases the coproduct is the respective direct sum (e.g. direct sum of vector bundles in the case of vector bundles).

Also:

### Vector bundles with external tensor product

We spell out in much detail the example of the category $Vect_{Set}$ of vector bundles over arbitrary base spaces and equipped with the external tensor product of vector bundles — for the simple special case that the base spaces are discrete topological spaces, i.e. plain sets:

###### Definition

For any ground field, write $Vect_{Set}$ for the category of indexed sets of vector spaces.

###### Remark

We may and will present objects $\mathcal{V}$ of $Vect_{Set}$ as pairs consisting of a set $S$ and a function $\mathcal{V}_{(-)}$ (really a functor on the discrete category on $S$) to Vect:

$\left( \array{ S &\longrightarrow& Vect \\ s &\mapsto& \mathcal{V}_s } \right) \;\; \in \;\; Vect_{Set} \mathrlap{\,.}$

###### Definition
$\boxtimes \,\colon\, Vect_{Set} \times VectSet \longrightarrow Vect_{Set}$

given by

$\big( \mathcal{V}_{(-)} \,\colon\, S \to \Vect \big) \,\boxtimes\, \big( \mathcal{V}'_{(-)} \,\colon\, S' \to \Vect \big) \;\; \coloneqq \;\; \left( \array{ S \times S' &\longrightarrow& \Vect \\ (s, s') &\mapsto& \mathcal{V}_s \otimes \mathcal{V}'_{s'} } \right) \mathrlap{\,.}$

###### Proposition
$\big( \mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to \Vect \big) \; \sqcup \; \big( \mathcal{V}^{(1)}_{(-)} \,\colon\, S_2 \to \Vect \big) \;\; \simeq \;\; \left( \array{ S_1 \sqcup S_2 &\longrightarrow& \Vect \\ s_i &\mapsto& \mathcal{V}^{(i)}_{s_i} } \right)$

###### Proof

It is immediate to check the universal property characterizing the coproduct.

###### Proposition

The external tensor product $\boxtimes$ (Def. ) distributes over the coproduct (Prop. ):

$\mathcal{E} \boxtimes ( \mathcal{V}^{(1)} \sqcup \mathcal{V}^{(2)}) \;\simeq\; \big( \mathcal{E} \boxtimes \mathcal{V}^{(1)} \big) \,\sqcup\, \big( \mathcal{E} \boxtimes \mathcal{V}^{(2)} \big)$

and hence gives a distributive monoidal category:

$\big( Vect_{Set} , \sqcup , \boxtimes \big) \;\in\; DistMonCat \mathrlap{\,.}$

###### Proof

Unwinding the above definitions and using that Set is a distributive category, we have the following sequence of natural isomorphisms:

$\begin{array}{l} \mathcal{E} \,\boxtimes\, \big( \mathcal{V}^{(1)} \,\sqcup\, \mathcal{V}^{(2)} \big) \\ \;\equiv\; \big( \mathcal{E}_{(-)} \,\colon\, S_E \to Vect \big) \,\boxtimes\, \Big( \big( \mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to Vect \big) \,\sqcup\, \big( \mathcal{V}^{(2)}_{(-)} \,\colon\, S_2 \to Vect \big) \Big) \\ \;\simeq\; \big( \mathcal{E}_{(-)} \,\colon\, S_E \to Vect \big) \,\boxtimes\, \left( \array{ S_1 \sqcup S_2 &\longrightarrow& Vect \\ s_i &\mapsto& \mathcal{V}^{(i)}_{s_i} } \right) \\ \;\simeq\; \left( \array{ S_E \times (S_1 \sqcup S_2) &\longrightarrow& Vect \\ (s_E , s_i) &\mapsto& \mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i} } \right) \\ \;\simeq\; \left( \array{ (S_E \times S_1) \,\sqcup\, (S_2 \times S_2) &\longrightarrow& Vect \\ (s_E , s_i) &\mapsto& \mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i} } \right) \\ \;\simeq\; \left( \array{ S_E \times S_1 &\longrightarrow& Vect \\ (s_E , s_1) &\mapsto& \mathcal{E}_{s_E} \otimes \mathcal{V}^{(1)}_{s_1} } \right) \,\sqcup\, \left( \array{ S_E \times S_2 &\longrightarrow& Vect \\ (s_E , s_2) &\mapsto& \mathcal{E}_{s_E} \otimes \mathcal{V}^{(2)}_{s_2} } \right) \\ \;\equiv\; \mathcal{E} \boxtimes \mathcal{V}^{(1)} \,\sqcup\, \mathcal{E} \boxtimes \mathcal{V}^{(2)} \end{array}$

## Properties

### The weakly semicartesian case

###### Remark

A monoidal category is finitary distributive if its tensor product preserves binary coproducts in each variable and the monoidal unit $I$ is weakly terminal (e.g., if there is a morphism $1 \to I$ out of a terminal object).

###### Proof

There is a canonical isomorphism

$x \otimes 0 + x \otimes y \to x \otimes (0 + y)$

and thus a canonical isomorphism

$\phi: x \otimes 0 + x \otimes y \to x \otimes y$

whose restriction along the coproduct inclusion $x \otimes y \to x \otimes 0 + x \otimes y$ is the identity $1_{x \otimes y}$. Let $k: x \otimes 0 \to x \otimes y$ be the restriction of $\phi$ along the other coproduct inclusion. Then $\phi$ induces an evident bijection

$\hom(x \otimes y, y) \stackrel{\langle [k], id \rangle}{\to} \hom(x \otimes 0, y) \times \hom(x \otimes y, y).$

Since $\hom(x \otimes y, y)$ is inhabited for all $x, y$ (with the help of some map $x \to I$, there is some map $x \otimes y \to I \otimes y \cong y$), this forces $\hom(x \otimes 0, y)$ to be a singleton for any $y$, so that $x \otimes 0$ is initial.

### Free monoids

A distinguishing feature of (infinitary) distributive monoidal categories is that the monad $T$ for monoid objects in such a category has a particularly simple expression:

$T X = \coprod_n X^{\otimes n}.$

The same is true for the monad on enriched graphs whose algebras are categories enriched over such a monoidal category. This also generalizes to lax monoidal categories, a.k.a. “multitensors”; see Weber 13.

## References

Last revised on February 8, 2024 at 03:43:35. See the history of this page for a list of all contributions to it.