English

What are symmetric monoidal categories?

Category Theory 2026-07-15 v1 Algebraic Geometry Algebraic Topology

Abstract

Symmetric monoidal categories have been understood since the 1960's and are central to many branches of mathematics. In particular, the construction of spectra from symmetric monoidal categories is at the heart of algebraic KK-theory. This construction starts from either categories with an action by a suitable operad \sP\sP or with suitable functors from the category \sF\sF of finite sets to the category Cat\mathbf{Cat} of categories. Infinite loop space theory, which codifies these constructions, led to the invention of \infty-categories. So why the title? We shall prove that the 22-category of symmetric monoidal categories is equivalent (in fact very nearly isomorphic) both to a 22-category of \sP\sP-pseudoalgebras and to an isomorphic 22-category of strictly special \sF\sF-pseudoalgebras. This equivalence underlies a streamlined equivariant and multiplicative enhancement of infinite loop space theory, but it should be of independent interest.

Cite

@article{arxiv.2607.13912,
  title  = {What are symmetric monoidal categories?},
  author = {Jiasen Liu and J. P. May and Kyle I. Roke and Hongyi Zhang and Keming Zhou},
  journal= {arXiv preprint arXiv:2607.13912},
  year   = {2026}
}