On the category of props

TL;DR

This paper introduces the category of props, an extension of operads and categories, highlighting its formal properties and establishing foundational tools for future research in algebraic structures.

Contribution

It develops a formal framework for props, including a free construction, multilinearity of functors, and connections to permutative categories and operads.

Findings

The category of props is bicomplete and symmetric monoidal.

A generalized free prop construction is defined using generalized graphs.

The relationship between props, operads, and permutative categories is established.

Abstract

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. The titular category has nice formal properties: it is bicomplete and is a symmetric monoidal category, with monoidal product closely related to the Boardman-Vogt tensor product of operads. Tools developed in this article, which is the first part of a larger work, include a generalized version of multilinearity of functors, a free prop construction defined on certain "generalized graphs", and the relationship between the category of props and the categories of permutative categories and of operads.