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

Abstract

Via the adjunction \( - *\mathbbm {1} \dashv \mathcal V(\mathbbm {1},-) :\textsf {Span}({\mathcal {V}}) \rightarrow {\mathcal {V}} \text {-} \textsf {Mat} \) and a cartesian monad T on an extensive category \( {\mathcal {V}} \) with finite limits, we construct an adjunction \( - *\mathbbm {1} \dashv {\mathcal {V}}(\mathbbm {1},-) :\textsf {Cat}(T,{\mathcal {V}}) \rightarrow ({\overline{T}}, \mathcal V)\text{- }\textsf{Cat} \) 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 \( - *\mathbbm {1} :\textsf {Set} \rightarrow {\mathcal {V}} \) 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.