distributive monoidal category

Distributive monoidal categories


Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Distributive monoidal categories


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


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 iY i i(XY i) X \otimes \coprod_i Y_i \cong \coprod_i (X\otimes Y_i)
iX iY i(X iY) \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 MM is the free monoidal category monad and MVVM V \xrightarrow{\otimes} V is the structure map of a monoidal category VV, then VV is distributive if it admits left Kan extensions along functors f:ABf:A\to B between discrete categories (of some size), and moreover if

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

is such a Kan extension, then so is

MA Mf MB Mϕ MV V.\array{ M A && \xrightarrow{M f} && M B\\ & \searrow & \xRightarrow{M \phi} & \swarrow\\ && M V\\ && \downarrow^\otimes\\ && V. }


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 last case).



The weakly semicartesian case


A monoidal category is finitary distributive if its tensor product preserves binary coproducts in each variable and the monoidal unit II is weakly terminal (e.g., if there is a morphism 1I1 \to I out of a terminal object).


There is a canonical isomorphism

x0+xyx(0+y)x \otimes 0 + x \otimes y \to x \otimes (0 + y)

and thus a canonical isomorphism

ϕ:x0+xyxy\phi: x \otimes 0 + x \otimes y \to x \otimes y

whose restriction along the coproduct inclusion xyx0+xyx \otimes y \to x \otimes 0 + x \otimes y is the identity 1 xy1_{x \otimes y}. Let k:x0xyk: x \otimes 0 \to x \otimes y be the restriction of ϕ\phi along the other coproduct inclusion. Then ϕ\phi induces an evident bijection

hom(xy,y)[k],idhom(x0,y)×hom(xy,y).\hom(x \otimes y, y) \stackrel{\langle [k], id \rangle}{\to} \hom(x \otimes 0, y) \times \hom(x \otimes y, y).

Since hom(xy,y)\hom(x \otimes y, y) is inhabited for all x,yx, y (with the help of some map xIx \to I, there is some map xyIyyx \otimes y \to I \otimes y \cong y), this forces hom(x0,y)\hom(x \otimes 0, y) to be a singleton for any yy, so that x0x \otimes 0 is initial.

Free monoids

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

TX= nX n. 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.


Last revised on January 8, 2021 at 22:44:20. See the history of this page for a list of all contributions to it.