Generalized Multicategories: Change-of-Base, Embedding, and Descent

Abstract

Via the adjunction -∗1⊣V(1,-):Span(V)→V-Mat and a cartesian monad T on an extensive category V with finite limits, we construct an adjunction -∗1⊣V(1,-):Cat(T,V)→(T¯,V)-Cat between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad T satisfies a suitable property, which holds for several examples. We verify, moreover, that the left adjoint is fully faithful, and preserves pullbacks, provided that the copower functor -∗1:Set→V is fully faithful. We also apply this result to study descent theory of generalized enriched multicategorical structures. These results are built upon the study of base-change for generalized multicategories, which, in turn, was carried out in the context of categories of horizontal lax algebras arising out of a monad in a suitable 2-category of pseudodouble categories.