nLab permutative category

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Category theory

Contents

Idea

A permutative category is a symmetric monoidal category (possibly taken to be internal to Top) in which associativity (including unitality) holds strictly. Also known as a symmetric strict monoidal category.

Definition

(May, def. 1) (Elmendorf-Mandell, def. 3.1).

A permutative category is a strict monoidal category equipped with a natural transformation B x,y:xyyxB_{x,y}:x \otimes y \rightarrow y \otimes x such that:

  • (B x,yId z);(Id yB x,z)=B x,yz(B_{x,y} \otimes Id_{z});(Id_{y} \otimes B_{x,z}) = B_{x,y \otimes z}
  • B x,y;B y,x=Id xyB_{x,y};B_{y,x} = Id_{x \otimes y}

In string diagrams:

Axioms of a permutative category using string diagrams

Properties

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.

Proposition

In a permutative category, for every object xx, we have B x,1=Id xB_{x,1}=Id_{x}.

Proof

Apply the two equations of the definition by putting y=1y=1 and z=1z=1. We obtain:

  • (B x,1) 2=B x,1(B_{x,1})^{2} = B_{x,1}
  • B x,1;B 1,x=Id xB_{x,1};B_{1,x} = Id_{x}

We obtain that B x,1=Id xB_{x,1}=Id_{x} by postcomposing the first equation by B 1,xB_{1,x}.

Note that it is not really possible to do this proof by using string diagrams.

The equation B I,x=Id xB_{I,x}=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 fId 1=f=Id 1ff \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 λ x:1xx\lambda_{x}:1 \otimes x \rightarrow x and ρ x:x1x\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).

References

An original account is in

  • John Isbell, On coherent algebras and strict algebras, J. Algebra 13 (1969)

Discussion in relation to symmetric spectra is in

Discussion in the context of K-theory of a permutative category is in

Discussion in the context of equivariant stable homotopy theory is in

Last revised on August 1, 2022 at 16:28:09. See the history of this page for a list of all contributions to it.