nLab commutative monoidal category



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



Commutative monoidal categories are symmetric monoidal categories whose tensor product is strictly associative and unital (as for permutative categories), but also strictly commutative, in that the associators, unitors, and braidings are all the identity natural transformation.

Notice that the coherence theorem for symmetric monoidal categories only says that symmetric monoidal categories are symmetric monoidally equivalent to strict monoidal categories whose braidings, however, may not be given by the identity.


Commutative monoidal categories are the natural type of category that Petri nets freely generate (Meseguer-Montanari 90, Baez-Master 18, Section 2).

The symmetric monoidal category of line bundles on a topological space, or smooth line bundles? on a manifold, or invertible sheaves? on a variety or scheme, is symmetric monoidally equivalent to a commutative monoidal category under the usual tensor product of these structures.

Any abelian group object in CatCat is a commutative monoidal category.


A commutative monoidal category is a commutative monoid object in Cat with its cartesian product. Equivalently, it is an internal category in the category of commutative monoids.

Explicitly, the data of a commutative monoidal category are:

  • A commutative monoid of objects C 1C_1,

  • a commutative monoid of morphisms C 0C_0,

  • source and target monoid homomorphisms s,t:C 1C 0s,t \colon C_1 \to C_0,

  • an identity assigning monoid homomorphism i:C 0C 1i \colon C_0 \to C_1 and,

  • a homomorphism :C 1× C0C 1C 1\circ \colon C_1 \times_C_0 C_1 \to C_1 representing composition.

These homorphisms are required to satisfy the axioms of a category. In particular, because composition is a commutative monoid homomorphism, it satisfies the interchange law

(gf)+(hk)=(g+h)(f+k) (g \circ f) + (h \circ k) = (g + h) \circ (f + k)

whenever all composites are defined.

Characterization up to equivalence

One might conjecture that a symmetric monoidal category is symmetric monoidally equivalent to a commutative monoidal category iff all its self-braidings

B x,x:xxxx B_{x,x} \colon x \otimes x \to x \otimes x

are identity morphisms. Note that since a commutative monoidal category has this property, and this property is invariant under symmetric-monoidal equivalence, the “only if” part of the characterization is certainly true.

This conjecture was proved in Kim 16 under the assumption that the objects of the monoidal category can be totally ordered (which is true, for instance, assuming the axiom of choice, or assuming the weaker ultrafilter principle which states that every filter extends to an ultrafilter).


Last revised on January 21, 2024 at 11:44:54. See the history of this page for a list of all contributions to it.