Monoidal Pull-Push II: Local Systems

TL;DR

This paper constructs an explicit lax monoidal functor for pull-push of local systems using Kan extensions, connecting spans of spaces to Cat_, and develops a framework of monoidal fibrations.

Contribution

It provides a simple, explicit construction of the pull-push functor as a lax monoidal functor via Kan extensions and introduces monoidal Beck-Chevalley fibrations.

Findings

Constructs pull-push of local systems as a lax monoidal functor.

Shows that horn filling problems can be solved using left Kan extensions.

Establishes a functor from spans of spaces to  with monoidal structure.

Abstract

Our aim in this work is to provide an explicit, simple construction of pull-push of local systems as a lax monoidal functor. To this end, we show that one can solve horn filling problems Cat_\infty using left Kan extensions, and use this to provide an explicit construction of a left Kan extension functor. We use this result to show that pull-push of local systems induces a functor from Span(S), the infinity category of spans of spaces, into Cat_\infty. We then develop a machinery of monoidal Beck-Chevalley fibrations, and use this to show that the pull-push functor above admits a lax monoidal structure.