category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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 $a \otimes b$ to $b \otimes a$ back to $a \otimes b$, 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 special case of the notion of symmetric pseudomonoid in a sylleptic monoidal 2-category.
A symmetric monoidal category is a braided monoidal category for which the braiding
satisfies the condition:
for all objects $x, y$
Intuitively this says that switching things twice in the same direction has no effect.
Expanding this out a bit: a symmetric monoidal category is, to begin with a category $M$ equipped with a functor
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 $SymmMonCat$ with:
This 2-category has (weak) 2-biproducts given by the cartesian product of underlying categories (analogously to how Ab has biproducts given by the cartesian product of underlying sets). For a proof, see Fong-Spivak, Theorem 2.3, or for a more abstract version involving pseudomonoids Schaeppi, Appendix A.
The group completion of 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
between the full subcategory of the stable homotopy category $Ho(Spectra)$ on the connective spectra and the homotopy category of $SymmMonCat$, regarded with the transferred structure of a category with weak equivalences.
This is due to (Thomason, 95). Further discussion is in (Mandell, 2010).
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 $SymmMonCat$ cannot carry a model category structure because it does not have all colimits. In some sense the “colimit completion” of $SymmMonCat$ 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.)
However, a subcategory of $SymmMonCat$ whose objects are Permutative categories and maps are strict symmetric monoidal functors, denoted by $Perm$ has a model category structure which is transferred from the natural model category structure on $Cat$, see Sharma. This model category structure is combinatorial, left-proper and a $Cat$-model category structure. It is referred to as the natural model category structure on $Perm$. The coherence theorem for symmetric monoidal categories states that each symmetric monoidal category is equivalent to a permutative category.
The aforementioned natural model category of permutative categories is NOT a symmetric monoidal closed model category. This shortcoming was overcome in [Sharma] by constructing a Quillen equivalent model category which is symmetric monoidal closed. A (unnormalized) $\Gamma$-category is a functor from $\Gamma^{op}$ to $Cat$, where $\Gamma^{op}$ is a skeletal category of finite based sets and based maps. The category of $\Gamma$-categories and natural transformations, denoted by $\Gamma$$Cat$, is a symmetric monoidal closed category under the Day convolution product. The aforementioned symmetric monoidal closed model category is constructed in Sharma as a left-Bousfield localization of the projective model category structure on $\Gamma$$Cat$. A $\Gamma$-category is fibrant in this model category if it satisfies the Segal’s condition in which case it is referred to as a coherently commutative monoidal category. The main result of Sharma is that an unnormalized version of the classical Segal’s nerve functor is the right Quillen functor of a Quillen equivalence between the natural model category of permutative categories and the symmetric monoidal closed model category of coherently commutative monoidal categories.
A symmetric monoidal category is equivalently a category that is equipped with the structure of an algebra over the little k-cubes operad for $k \geq 3$
Details are in examples 1.2.3 and 1.2.4 of
Tannaka duality for categories of modules over monoids/associative algebras
monoid/associative algebra | category of modules |
---|---|
$A$ | $Mod_A$ |
$R$-algebra | $Mod_R$-2-module |
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 |
2-Tannaka duality for module categories over monoidal categories
monoidal category | 2-category of module categories |
---|---|
$A$ | $Mod_A$ |
$R$-2-algebra | $Mod_R$-3-module |
Hopf monoidal category | monoidal 2-category (with some duality and strictness structure) |
3-Tannaka duality for module 2-categories over monoidal 2-categories
monoidal 2-category | 3-category of module 2-categories |
---|---|
$A$ | $Mod_A$ |
$R$-3-algebra | $Mod_R$-4-module |
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.
The internal logic of (closed) symmetric monoidal categories is called linear logic. This notably contains quantum logic.
Every cartesian monoidal category is necessarily symmetric monoidal, due to the essential uniqueness of the categorical product. This includes cases such as Set, Cat.
For $k$ some field, the category Vect of $k$-vector spaces carries the standard structure of a monoidal category coming from the tensor product, over $k$, of vector spaces. The standard braiding that identifies $V \otimes W$ with $W \otimes V$ by mapping homogeneous elements $v \otimes w$ to $w \otimes v$ obviously makes Vect into a symmetric monoidal category.
The category of $\mathbb{Z}_2$-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 $v$ and $w$ are passed past each other : $v \otimes w \mapsto - w \otimes v$. This non-trivial symmetric monoidal structure on $Vect[\mathbb{Z}_2]$ defines the symmetric monoidal category of super vector spaces.
The monoidal category of graded modules over a commutative ring (with the usual tensor product of graded modules) can be made into a braided monoidal category with the braiding
The braiding $x \otimes y \mapsto (-1)^{|x| |y|} y \otimes x$ (where $|x|$ and $|y|$ denote the degrees) is also commonly used.
More generally, for any invertible element $u$ of the base ring, there is the braiding $x \otimes y \mapsto u^{|x| |y|} y \otimes x$, and these braidings are the only possible. The resulting braided monoidal category is symmetric if and only if $u^2 = 1$.
symmetric monoidal category, symmetric monoidal (∞,1)-category, symmetric monoidal (∞,n)-category
Exposition of basics of monoidal categories and categorical algebra:
geometry of physics – categories and toposes, Section 2: Basic notions of categorical algebra
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, chapter 2 of Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf)
A survey of definitions of symmetric monoidal categories, symmetric monoidal functors and symmetric monoidal natural transformations, is also in
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
Brendan Fong and David I, Spivak, Supplying bells and whistles in symmetric monoidal categories, arxiv, 2019
Daniel Schäppi, Ind-abelian categories and quasi-coherent sheaves, arxiv, 2014
Amit Sharma, Symmetric monoidal categories and $\Gamma$-categories Theory and applications of Categories, Vol. 35, No. 14, (2020) pp. 417-512