With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Generalizing the classical notion of commutative monoid, one can define a commutative monoid (or commutative monoid object) in any symmetric monoidal category . These are monoids in a monoidal category whose multiplicative operation is commutative. Classical commutative monoids are of course just commutative monoids in Set with the cartesian product.
Given a monoidal category , then a monoid internal to is
such that
(associativity) the following diagram commutes
where is the associator isomorphism of ;
(unitality) the following diagram commutes:
where and are the left and right unitor isomorphisms of .
Moreover, if has the structure of a symmetric monoidal category with symmetric braiding , then a monoid as above is called a commutative monoid in if in addition
(commutativity) the following diagram commutes
A homomorphism of monoids is a morphism
in , such that the following two diagrams commute
and
Write for the category of monoids in , and for its subcategory of commutative monoids.
Write for the category Ab of abelian groups, equipped with the tensor product of abelian groups whose tensor unit is the additive group of integers. With the evident braiding this is a symmetric monoidal category.
A commutative monoid in is equivalently a commutative ring.
(differential graded-commutative algebras)
The category of chain complexes with its tensor product of chain complexes carries a symmetric monoidal braiding given on elements in definite degree by
The corresponding commutative monoid objects are the differential graded-commutative algebras.
(differential graded-commutative superalgebras)
The category of chain complexes of super vector spaces with its tensor product of chain complexes carries two symmetric monoidal braidings given on elements in definite bidegree by
;
.
The corresponding commutative monoid objects are the differential graded-commutative superalgebras.
sign rule for differential graded-commutative superalgebras
(different but equivalent)
Deligneβs convention | Bernsteinβs convention | |
---|---|---|
common in discussion of | supergravity | AKSZ sigma-models |
representative references | Bonora et. al 87, Castellani-DβAuria-FrΓ© 91, Deligne-Freed 99 | AKSZ 95, Carchedi-Roytenberg 12 |
Since the two braidings above are equivalent (this Prop) the corresponding two categories of differential graded-commutative superalgebras are also canonically equivalence of categories:
(commutative ring spectra, E-infinity rings)
Write and and for the categories, respectively of symmetric spectra, orthogonal spectra and pre-excisive functors, equipped with their symmetric monoidal smash product of spectra, whose tensor unit is the corresponding standard incarnation of the sphere spectrum.
A commutative monoid in any one of these three categories is equivalently a commutative ring spectrum in the strong sense: via the respective model structure on spectra it represents an E-infinity ring.
(in a cocartesian monoidal category)
Every object in a cocartesian monoidal category becomes a commutative monoid in a unique way: the multiplication must be the fold map , and the counit must be the unique map . Similarly every morphism in becomes a morphism of commutative monoid objects, so the category of commutative monoid objects in is isomorphic to .
(in )
Since the category of commutative monoids (in ) is cocartesian, the category of commutative monoids in is again . Finite coproducts of commutative monoids are also finite products, so the category of commutative monoids in is also .
Categorical properties of commutative monoid objects in symmetric monoidal categories are spelled out in sections 1.2 and 1.3 of
A summary is in section 4.1 of
See also MO/180673.
Last revised on May 14, 2022 at 15:52:41. See the history of this page for a list of all contributions to it.