Categorical and algebraic models for path

Plática dada por Manuel Rivera (University of Miami-Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, CINVESTAV) en el evento Algebraic Geometry in 2017 el sábado 2 de diciembre del 2017 en Puerto Escondido, Oaxaca, México

In this talk I will describe how the following three functors are related:

1) the path space functor and its relatives (based path space, based loop space, free loop space, etc…)
2) the cobar functor from the category of differential graded coalgebras to the category of differential graded algebras
3) the rigidification functor from simplicial sets to simplicial categories

1) is a classical and important construction which appears all over through geometry and topology
2) is a purely algebraic construction introduced by Frank Adams in the 1950’s to obtain an algebraic model for the based loop space of a simply connected space which is suitable for computations.
3) was introduced by J. Lurie and A. Joyal in order to compare different models for infinity categories. The key to relate these three functors is to introduce a cubical version of 3). I will discuss some consequences that follow from understanding how these functors are related: we use these ideas to construct small combinatorial and algebraic models for the based paths and free loops fibrations over a connected but possibly non-simply connected space generalizing Adams’ classical result, we obtain a strict adjoint for the differential graded nerve functor, and we may formulate a transparent algebraic definition for infinity local systems.

This is joint work with M. Zeinalian (arXiv:1612.04801) and S.Saneblidze (arXiv:1706.00983).

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Departamento de Matemáticas del CINVESTAV

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