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
A permutative category is a symmetric monoidal category (possibly taken to be internal to Top) in which associativity and unitality hold strictly. Also known as a symmetric strict monoidal category.
(May, def. 1) (Elmendorf-Mandell, def. 3.1).
A permutative category is a strict monoidal category equipped with a natural transformation $B_{x,y}:x \otimes y \rightarrow y \otimes x$ such that:
In string diagrams:
Every symmetric monoidal category is equivalent to a permutative one (Isbell).
The nerve of a permutative category is an E-infinity space, and therefore can be infinitely delooped to obtain an infinite loop space as its group completion.
In a permutative category, for every object $x$, we have $B_{x,1}=Id_{x}$.
Apply the two equations of the definition by putting $y=1$ and $z=1$. We obtain:
We obtain that $B_{x,1}=Id_{x}$ by postcomposing the first equation by $B_{1,x}$.
Note that it is not really possible to do this proof by using string diagrams.
The equation $B_{x,1}=Id_{x}$ is taken as an axiom of a permutative category in the references above. This is maybe the consequence of a lack of care about what is an identity natural transformation in the definition of a strict monoidal category. The equations $f \otimes Id_{1} = f = Id_{1} \otimes f$ are of critical importance in the proposition above and they are obtained by requiring that the structural natural isomorphims $\lambda_{x}:1 \otimes x \rightarrow x$ and $\rho_{x}:x \otimes 1 \rightarrow x$ are identity natural transformations. However, even knowing this, the proposition is not completely trivial and appears in a version for non-strict braided monoidal categories in the paper “Braided monoidal categories” (Joyal, Street, 1986).
An original account is in
Discussion in relation to symmetric spectra is in
Discussion in the context of K-theory of a permutative category is in
Peter May, The spectra associated to permutative categories, Topology 17 (1978) (pdf)
Peter May, $E_\infty$-spaces, group completions, and permutative categories, London Math. Soc. Lecture Notes No. 11, 1974, 61-94 (doi:10.1017/CBO9780511662607.008, pdf)
Peter May, $E_\infty$ Ring Spaces and $E_\infty$ Ring spectra, Springer lectures notes in mathematics, Vol. 533, (1977) (pdf) chaper VI
Anthony Elmendorf, Michael Mandell, Rings, modules and algebras in infinite loop space theory, K-Theory 0680 (web, pdf)
Discussion in the context of equivariant stable homotopy theory is in
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