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Macquarie University  Department of Mathematics

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Categories in Algebra, Geometry and Mathematical Physics

Conference in honour of Ross Street's sixtieth birthday

July 11-16 2005, Macquarie University, Sydney

Wed, 13 July:  10:45 - 11:25

Bimodules over operads: encoding deep structure of morphisms
Scott, Jonathan (École Polytechnique Fédèrale de Lausanne)

\thanks{Joint work with K.\ Hess (EPFL) and P.-E.\ Parent (U. Ottawa).}
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An operad $P$ determines a category of $P$-algebras and their morphisms. The algebras and morphisms are linked, in the sense that in order to alter the morphisms, one must alter $P$, changing the algebras in the process.

We will show how $P$-bimodules can be used to give more freedom in describing morphisms. Let $R$ be a $P$-bimodule. If $R$ is a $P$-cooperad, then the category of $P$-algebras and ``$R$-relative" morphisms is a full subcategory of the Kleisli category for the triple associated to $R$. We apply this idea to unravel the algebraic structure in morphisms of bar/cobar constructions.

As a concrete example, we construct a bimodule over the associative operad of chain complexes that yields the category of associative algebras and ``strongly homotopy-multiplicative" morphisms.

Typeset PDF of this abstract.