nLab distributive monoidal category

Distributive monoidal categories

Context

Monoidal categories

monoidal categories

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. Thus, we have canonical isomorphisms

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

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

A distributive cartesian monoidal category is a distributive category, while any distributive monoidal category is a particular case of a rig category. See distributivity for monoidal structures.

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

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 last case).

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).

• Mark Weber, Multitensors and monads on categories of enriched graphs, spnet