广义多范畴：基变换、嵌入与下降
Generalized multicategories: change-of-base, embedding, and descent
AI总结：
本文在广义多范畴之间构造并研究了一个伴随对，证明了左伴随的完全忠实性与拉回保持性，并将其应用于广义富集多范畴结构的下降理论。
摘要：
通过伴随对 $- \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Span}(\mathcal V) \to \mathcal V \text{-} \mathsf{Mat}$ 以及一个在具有有限极限的外延范畴 $\mathcal V$ 上的笛卡尔单子 $T$，我们构造了一个伴随对 $- \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Cat}(T,\mathcal V) \to (\overline T, \mathcal V)\text{-}\mathsf{Cat}$，它位于广义富集多范畴的范畴与广义内部多范畴的范畴之间，条件是单子 $T$ 满足一个适当的条件，而该条件被若干例子所满足。此外，我们验证了左伴随是完全忠实的，并且在余幂函子 $- \boldsymbol{\cdot} 1 \colon \mathsf{Set} \to \mathcal V$ 是完全忠实的条件下保持拉回。我们还将这一结果应用于研究广义富集多范畴结构的下降理论。这些结果建立在广义多范畴的基变换研究之上，而后者又是在由伪双范畴的适当 2-范畴中的单子所产生的水平松弛代数范畴的背景下进行的。
英文摘要：
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.
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