Modules and Morita theorem for operads
type: Item
instance of: scholarly article
title: Modules and Morita theorem for operads
zbMATH Open document ID: 1001.18004
author: Mikhail Kapranov
author: Yuri I. Manin
published in: American Journal of Mathematics
publication date: 2002-12-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9906063
full work available at URL: ajm123.5.html
Mathematics Subject Classification ID: 18D50
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 55P48
Mathematics Subject Classification ID: 58A50
zbMATH DE Number: 1688820
zbMATH Keywords: operads
zbMATH Keywords: modules over operads
zbMATH Keywords: Morita theory
zbMATH Keywords: plethysm tensor product
MaRDI profile type: Publication
OpenAlex ID: W2048763806
arXiv ID: math/9906063
DOI: 10.1353/AJM.2001.0033
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review text: In the article the authors investigate generalisations of classical Morita theory in the context of linear operads. For a \(k\)-linear operad \(\mathcal P\) matrix operads \(\text{Mat}(n,{\mathcal P})\) have been be constructed, and it has been shown that their categories of representations are equivalent.NEWLINENEWLINENEWLINEA \(k\)-linear operad can be defined as a monoid in the category of \({\mathcal S}\)-spaces, \(({{\mathcal S}}\)-\({\mathcal V}ect,\circ)\), where ``\(\circ\)'' is the so-called plethysm tensor product and \({\mathcal S}\) is the category of standard finite sets and their bijections. The plethysm product, however, is not symmetric. In particular the notions of left and right modules over an operad get asymmetric. Besides the plethysm product two other important tensor products can be defined on \({{\mathcal S}}\)-\({\mathcal V}ect\). And one of these, call it \(\otimes\), corresponds to composition of formal power series, whereas the plethysm is an analogue of multiplication of power series.NEWLINENEWLINENEWLINEIn order to construct a Morita theory for operads the authors work with right \(\mathcal P\)-modules and replace left modules by \(\mathcal P\)-algebras as follows. The relative plethysm product for modules over operads will be defined in a way similar to the usual tensor product of a right and a left module over an algebra or over a monoid in a monoidal category. Furthermore for two operads \(\mathcal P\) and \(\mathcal Q\) a \(({\mathcal Q},{\mathcal P})\)-bimodule can be defined. Then a canonical associativity isomorphism can be obtained for the relative plethysm product NEWLINE\[NEWLINE(L\circ_{\mathcal Q} M)\circ_{\mathcal P} N \simeq L\circ_{\mathcal Q}(M\circ_{\mathcal P} N)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE(L\circ_{\mathcal Q} M)\circ_{\mathcal P}\langle A\rangle \simeq L\circ_{\mathcal Q}\langle M\circ_{\mathcal P} A\rangleNEWLINE\]NEWLINE for \(L\) a left \(\mathcal Q\)-module, \(N\) a right \(\mathcal P\)-module, \(M\) a \(…
reviewed by: Bernhard Drabant
description: scientific article; zbMATH DE number 1688820