Generalized multicategories: change-of-base, embedding, and descent

Abstract

Via the adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Span}(\mathcal V) \to \mathcal V \text{-} \mathsf{Mat} $ and a cartesian monad $ T $ on an extensive category $ \mathcal V $ with finite limits, we construct an adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Cat}(T,\mathcal V) \to (\overline T, \mathcal V)\text{-}\mathsf{Cat} $ between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad $ T $ satisfies a suitable condition, which is satisfied by several examples. We verify, moreover, that the left adjoint is fully faithful, and preserves pullbacks, provided that the copower functor $ - \boldsymbol{\cdot} 1 \colon \mathsf{Set} \to \mathcal 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.