nLab distributive monoidal category

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

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

• Ab, the category of abelian groups equipped with the tensor product of abelian groups

• $R$Mod, the category of modules over a commutative ring $R$, equipped with the tensor product of modules

• $k$Vect = $k$Mod, the category of vector space over some field $k$, equipped with the tensor product of vector spaces,

• $k$Vect(X), the category of (topological) vector bundles for $X$ some (topological) space, equipped with the tensor product of vector bundles.

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

Also:

• the category of pointed topological spaces with respect to forming smash product and wedge sum (e.g. Hatcher, Section 4.F).

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.

Related concepts

• distributive law

• distributivity for monoidal structures

  • distributive monoidal category

  • bipermutative category

  • linearly distributive category

  • rig category

References

• Mark Weber, Multitensors and monads on categories of enriched graphs, TAC 28(26), 2013, (tac:28-26)