What are symmetric monoidal categories?
arXiv:2607.13912
The paper shows that the 2‑category of symmetric monoidal categories is essentially equivalent to the 2‑category of pseudoalgebras for a suitable operad and to the 2‑category of strictly special pseudoalgebras over the category of finite sets, providing a categorical foundation for equivariant and multiplicative infinite loop space theory.
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 -theory. This construction starts from either categories with an action by a suitable operad $\sP$ or with suitable functors from the category $\sF$ of finite sets to the category of categories. Infinite loop space theory, which codifies these constructions, led to the invention of -categories. So why the title? We shall prove that the -category of symmetric monoidal categories is equivalent (in fact very nearly isomorphic) both to a -category of $\sP$-pseudoalgebras and to an isomorphic -category of strictly special $\sF$-pseudoalgebras. This equivalence underlies a streamlined equivariant and multiplicative enhancement of infinite loop space theory, but it should be of independent interest.