nLab distributive monoidal category

For the cartesian case see at distributive 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: i.e. such that the canonical morphisms

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

$$\coprod_i (X_i\otimes Y)\to \Big(\coprod_i X_i\Big) \otimes Y$$

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.

See also at 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

Various

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 over 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 case of vector bundles).

Also:

• the category of pointed sets with respect to forming smash product and wedge sum,

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

• $k$VectBund, the category of (topological) vector bundles over any (convenient) topological spaces, equipped with the external tensor product of vector bundles (more on this below).

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

The “external” tensor product on $Vect_{Set}$ is the functor

$$\boxtimes \,\colon\, Vect_{Set} \times Vect_{Set} \longrightarrow Vect_{Set}$$

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

The coproduct in $Vect_{Set}$ is given by disjoint union of bundles:

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

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.

Related concepts

• distributive law

• distributivity for monoidal structures

  • distributive monoidal category

  • bipermutative category

  • linearly distributive category

  • rig category

References

• Hans-Joachim Baues, Mamuka Jibladze, Andy Tonks, first page of: Cohomology of monoids in monoidal categories, in: Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202, AMS (1997) 137-166 [doi:10.1090/conm/202, preprint:pdf]

• Anna Labella, Categories with sums and right distributive tensor product, Journal of Pure and Applied Algebra 178 3 (2003) 273-296 [doi:10.1016/S0022-4049(02)00169-X]

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