Labelled cospan categories and properads

TL;DR

This paper proves a conjecture establishing an equivalence between properads and labelled cospan categories, providing a strict categorical framework and advancing the understanding of their structural relationship.

Contribution

It demonstrates a strict categorical equivalence between properads and labelled cospan categories, confirming Steinebrunner's conjecture with a new proof using the symmetric monoidal envelope functor.

Findings

Properads are equivalent to strict labelled cospan categories.

The symmetric monoidal envelope functor provides the key to the equivalence.

The work confirms a conjecture in the categorical theory of algebraic structures.

Abstract

We prove Steinebrunner's conjecture on the biequivalence between (colored) properads and labelled cospan categories. The main part of the work is to establish a 1-categorical, strict version of the conjecture, showing that the category of properads is equivalent to a category of strict labelled cospan categories via the symmetric monoidal envelope functor.