nLab distributive monoidal category

Distributive monoidal categories

For the cartesian case see at distributive category.


Monoidal categories

monoidal categories

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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: i.e. such that the canonical morphisms

i(XY i)X iY i\coprod_i (X\otimes Y_i)\to X \otimes \coprod_i Y_i
i(X iY) iX iY\coprod_i (X_i\otimes Y)\to \coprod_i X_i \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 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 case of vector bundles).


Vector bundles with external tensor product

We spell out in much detail the example of the category Vect Set 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:


For any ground field, write Vect Set Vect_{Set} for the category of indexed sets of vector spaces.


We may and will present objects 𝒱\mathcal{V} of Vect Set Vect_{Set} as pairs consisting of a set SS and a function 𝒱 ()\mathcal{V}_{(-)} (really a functor on the discrete category on SS) to Vect:

(S Vect s 𝒱 s)Vect Set. \left( \array{ S &\longrightarrow& Vect \\ s &\mapsto& \mathcal{V}_s } \right) \;\; \in \;\; Vect_{Set} \mathrlap{\,.}


The “externaltensor product on Vect Set Vect_{Set} is the functor

:Vect Set×VectSetVect Set \boxtimes \,\colon\, Vect_{Set} \times VectSet \longrightarrow Vect_{Set}

given by

(𝒱 ():SVect)(𝒱 ():SVect)(S×S Vect (s,s) 𝒱 s𝒱 s). \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{\,.}


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

(𝒱 () (1):S 1Vect)(𝒱 () (1):S 2Vect)(S 1S 2 Vect s i 𝒱 s i (i)) \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)


It is immediate to check the universal property characterizing the coproduct.


The external tensor product \boxtimes (Def. ) distributes over the coproduct (Prop. ):

(𝒱 (1)𝒱 (2))(𝒱 (1))(𝒱 (2)) \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:

(Vect Set,,)DistMonCat. \big( Vect_{Set} , \sqcup , \boxtimes \big) \;\in\; DistMonCat \mathrlap{\,.}


Unwinding the above definitions and using that Set is a distributive category, we have the following sequence of natural isomorphisms:

(𝒱 (1)𝒱 (2)) ( ():S EVect)((𝒱 () (1):S 1Vect)(𝒱 () (2):S 2Vect)) ( ():S EVect)(S 1S 2 Vect s i 𝒱 s i (i)) (S E×(S 1S 2) Vect (s E,s i) s E𝒱 s i (i)) ((S E×S 1)(S 2×S 2) Vect (s E,s i) s E𝒱 s i (i)) (S E×S 1 Vect (s E,s 1) s E𝒱 s 1 (1))(S E×S 2 Vect (s E,s 2) s E𝒱 s 2 (2)) 𝒱 (1)𝒱 (2) \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}


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 February 8, 2024 at 03:43:35. See the history of this page for a list of all contributions to it.