nLab commutative monoidal category

Idea

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.

Examples

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 $Cat$ is a commutative monoidal category.

Definition

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 $C_1$,

• a commutative monoid of morphisms $C_0$,

• source and target monoid homomorphisms $s,t \colon C_1 \to C_0$,

• an identity assigning monoid homomorphism $i \colon C_0 \to C_1$ and,

• a homomorphism $\circ \colon C_1 \times_C_0 C_1 \to C_1$ 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

$$(g \circ f) + (h \circ k) = (g + h) \circ (f + k)$$

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

$$B_{x,x} \colon x \otimes x \to x \otimes x$$

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).

References

• José Meseguer, Ugo Montanari?, Petri Nets are Monoids (pdf)

• John Baez, Jade Master, Section 2 of: Open Petri Nets (arXiv:1808.05415)

• James Dolan, Doctrines of Algebraic Geometry

• Youngsoo Kim?, A Note on Strict Commutativity of a Monoidal Product, Pure and Applied Mathematics Journal Volume 5, Issue 5, October 2016, pp. 155-159, url