category with duals (list of them)
dualizable object (what they have)
A symmetric monoidal category is a category with a product operation – a monoidal category – for which the product is as commutative as possible.
The point is that there are different degrees to which higher categorical products may be commutative. While a bare monoid is either commutative or not, a monoidal category may be a braided monoidal category – which already means that the order of products may be reversed up to some isomorphism – without being symmetric monoidal – which means that changing the order of a product twice, from to back to , indeed does yield a result equal to the original.
For higher monoidal categories there are accordingly ever more shades of the notion of “commutativity” of the monoidal product. This is described in detail at k-tuply monoidal n-category.
In general, the term symmetric monoidal is used for the maximally commutative case. See for instance symmetric monoidal (∞,1)-category. Notably, a symmetric monoidal ∞-groupoid is, under the homotopy hypothesis, the same as a connective spectrum.
A symmetric monoidal category is a braided monoidal category for which the braiding
satisfies an additional axiom:
for all objects . Intuitively this says that switching things twice has no effect.
called the tensor product, an object
called the unit object, a natural isomorphism
called the associator, a natural isomorphism
called the left unitor, a natural isomorphism
called the right unitor, and a natural isomorphism
called the braiding. We then demand that the associator obey the pentagon identity, which says this diagram commutes:
We demand that the associator and unitors obey the triangle identity, which says this diagram commutes:
We demand that the braiding and associator obey the first hexagon identity:
And lastly, we demand that
(The definition of braided monoidal category has two hexagon identities, but either one implies the other given this equation.)
There is a strict 2-category with:
The nerve of a symmetric monoidal category is always an infinite loop space, hence the degree-0-space of a connective spectrum. One calls this also the K-theory spectrum of the symmetric monoidal category:
This construction extended to an equivalence of categories
Notice that this is almost the complete analog in stable homotopy theory of the Quillen equivalence between the Thomason model structure on Cat and the standard model structure on simplicial sets. Only that cannot carry a model category structure because it does not have all colimits. In some sense the “colimit completion” of is the category of multicategories. Once expects that this carries a model structure that refines the above equivalence of homotopy categories to a Quillen equivalence.
(This is currently being investigated by Elmendorf, Nikolaus and maybe others.)
Details are in examples 1.2.3 and 1.2.4 of
|monoid/associative algebra||category of modules|
|sesquialgebra||2-ring = monoidal presentable category with colimit-preserving tensor product|
|bialgebra||strict 2-ring: monoidal category with fiber functor|
|Hopf algebra||rigid monoidal category with fiber functor|
|hopfish algebra (correct version)||rigid monoidal category (without fiber functor)|
|weak Hopf algebra||fusion category with generalized fiber functor|
|quasitriangular bialgebra||braided monoidal category with fiber functor|
|triangular bialgebra||symmetric monoidal category with fiber functor|
|quasitriangular Hopf algebra (quantum group)||rigid braided monoidal category with fiber functor|
|triangular Hopf algebra||rigid symmetric monoidal category with fiber functor|
|supercommutative Hopf algebra (supergroup)||rigid symmetric monoidal category with fiber functor and Schur smallness|
|form Drinfeld double||form Drinfeld center|
|trialgebra||Hopf monoidal category|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|
The Grothendieck group of a monoidal category naturally has the structure of a monoid, of an abelian monoidal category that of a ring, of an abelian braided monoidal category that of a commutative ring and, finally, of an abelian symmetric monoidal category that of a Lambda-ring. See there for more.
For some field, the category Vect of -vector spaces carries the standard structure of a monoidal category coming from the tensor product, over , of vector spaces. The standard braiding that identifies with by mapping homogeneous elements to obviously makes Vect into a symmetric monoidal category.
The category of -graded vector spaces, on the other hand, has two different symmetric monoidal extensions of the standard tensor product monoidal structure. One is the trivial one from above, the other is the one that induces a a sign when two odd-graded vectors and are passed past each other : . This non-trivial symmetric monoidal structure on defines the symmetric monoidal category of super vector spaces.
For an elementary introduction to symmetric monoidal categories using string diagrams, see:
The theorem that symmetric monoidal categories model all connective spectra is due to
More discussion is in