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The circle product of \(\mathcal O\)-bimodules with \(\mathcal O\)-algebras, with applications.

Banff International Research Station for Mathematical Innovation and Discovery

If \(\mathcal O\) is an operad (in \(S\)-modules for example), \(M\) is an \(\mathcal O\)-bimodule, and \(A\) is an \(\mathcal O\)-algebra, then \(M \circ_{\mathcal O}A\) is again an \(\mathcal O\)-algebra. A bar construction model for a derived version of this has been studied in joint work with Luis Pereira. Then one can take advantage of the fact that this construction is very friendly in the first variable to both do homotopical analysis and to find extra structure. In particular, with \(A\) an augmented commutative \(S\)-algebra, our approach allows one to define the augmentation ideal filtration of \(A\) together with composition structure (as needed in my recent work on Hurewicz maps for infinite loopspaces), and also recover the filtration I found a decade ago on the tensor product \(K \otimes A\), with \(K\) is a based space (implicitly appearing in ongoing work by Behrens and Rezk).

Mathematics; Algebraic topology; Category theory; homological algebra

video/mp4

Author affiliation: University of Virginia

2016-11-24

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59796

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/