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

TL;DR

This paper develops a framework connecting generalized multicategories and internal multicategories via adjunctions, with applications to descent theory, under certain conditions on monads and copower functors.

Contribution

It constructs an adjunction between categories of generalized and internal multicategories using change-of-base techniques, extending the theory of multicategories in a broad categorical context.

Findings

The left adjoint is fully faithful and preserves pullbacks under certain conditions.

The framework applies to various examples satisfying the monad conditions.

Provides new insights into descent theory for generalized multicategorical structures.

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.