# Monoidal structures on the categories of quadratic data

## TL;DR

This paper explores the construction of 2-monoidal structures on categories of quadratic algebras and operads, extending Vallette's framework and inspired by Manin's observations, to better understand operadic and algebraic structures in topology and quantum algebra.

## Contribution

It provides a detailed exposition and generalization of 2-monoidal structures on quadratic algebras and operads, inspired by Manin's remark, enhancing the understanding of operadic structures in algebra and topology.

## Key findings

- Constructed 2-monoidal structures on quadratic algebras and operads.
- Extended Vallette's 2-monoidal framework to quadratic contexts.
- Potentially improves understanding of operads in topology and quantum algebra.

## Abstract

The notion of 2--monoidal category used here was introduced by B.~Vallette in 2007 for applications in the operadic context. The starting point for this article was a remark by Yu. Manin that in the category of quadratic algebras (that is, "quantum linear spaces") one can also define 2--monoidal structure(s) with rather unusual properties. Here we give a detailed exposition of these constructions, together with their generalisations to the case of quadratic operads. Their parallel exposition was motivated by the following remark. Several important operads/cooperads such as genus zero quantum cohomology operad, the operad classifying Gerstenhaber algebras, and more generally, (co)operads of homology/cohomology of some topological operads, start with collections of quadratic algebras/coalgebras rather than simply linear spaces. Suggested here enrichments of the categories to which components of these operads belong, as well of the operadic structures themselves, might lead to the better understanding of these fundamental objects. New version includes minor editorial changes and new references.