Title: Rectification of homotopy coherent algebraic structures to strict ones
Speaker: Dmitri Pavlov, Göttingen
Abstract: Why are homotopy coherent associative (i.e., A-infinity) monoids in simplicial sets always equivalent to strictly associative ones, whereas homotopy coherent commutative (i.e., E-infinity) monoids are not? In this talk I will discuss a necessary and sufficient criterion that allows one to answer this and many similar questions about rectification of homotopy coherent structures (specifically, those given by operads) to strict ones with relative ease. Apart from simplicial sets, the underlying category can also be the category of topological spaces, chain complexes, simplicial presheaves, and more generally, any model category satisfying some mild additional conditions. Of particular interest is the case of algebraic structures on structured spectra, for example, symmetric spectra, for which concrete applications force us to consider symmetric spectra in
categories more general than simplicial sets, for example, motivic spaces of Morel and Voevodsky.
It turns out that the criterion mentioned above is automatically verified under some mild conditions on the underlying category (satisfied, for example, by simplicial sets and motivic spaces), which means that homotopy coherent algebraic structures in symmetric spectra can always be rectified to strict ones. If time permits, we will discuss applications to Deligne cohomology, Toën-Vezzosi homotopical algebraic geometry, Goerss-Hopkins obstruction theory, enriched categories, operads, factorization algebras. Joint work with Jakob Scholbach (Münster).
