# Monoidal Structures on the Categories of Quadratic Data

### Yuri Ivanovich Manin

Max-Planck-Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany### Bruno Vallette

Laboratoire Analyse, Géométrie et Applications, Université Sorbonne Paris Nord, CNRS, UMR 7539, 93430 Villetaneuse, France

## Abstract

The notion of 2-monoidal category used here was introduced by *B. Vallette* in [J. Pure Appl. Algebra 208, No. 2, 699--725 (2007; Zbl 1109.18002); Trans. Am. Math. Soc. 359, No. 10, 4865--4943 (2007; Zbl 1140.18006)] 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 better understanding of these fundamental objects.

## Cite this article

Yuri Ivanovich Manin, Bruno Vallette, Monoidal Structures on the Categories of Quadratic Data. Doc. Math. 25 (2020), pp. 1727–1786

DOI 10.4171/DM/784