distributive monoidal category


Monoidal categories

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. }


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).


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

Last revised on May 26, 2017 at 01:45:31. See the history of this page for a list of all contributions to it.