From the Sigma-type to the Grothendieck construction

Abstract

We translate properties of the Sigma-type in Martin-L\"of Type Theory (MLTT) to properties of the Grothendieck construction in category theory. Namely, equivalences in MLTT that involve the Sigma-type motivate isomorphisms between corresponding categories that involve the Grothendieck construction. The type-theoretic axiom of choice and the "associativity" of the Sigma-type are the main examples of this phenomenon that are treated here.