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

• DOI: 10.1007/s10485-024-09775-y
• Publication Date: 2024-10-30T16:17:10Z
• Authors: Rui Prezado; Fernando Lucatell...
• Keywords: Descent (aeronautics); Base (topology); Theory of computation

Abstract Via the adjunction $$ - *\mathbbm {1} \dashv \mathcal V(\mathbbm {1},-) :\textsf {Span}({\mathcal {V}}) \rightarrow {\mathcal {V}} \text {-} \textsf {Mat} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mn>1</mml:mn> <mml:mo>⊣</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mo>,</mml:mo> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>Span</mml:mi> <mml:mo>→</mml:mo> <mml:mtext>-</mml:mtext> <mml:mi>Mat</mml:mi> </mml:mrow> </mml:math> and a cartesian monad T on an extensive category with finite limits, we construct {V}}(\mathbbm {Cat}(T,{\mathcal ({\overline{T}}, V)\text{- }\textsf{Cat} <mml:mi>Cat</mml:mi> <mml:mi>T</mml:mi> <mml:mover> <mml:mo>¯</mml:mo> </mml:mover> <mml:mspace/> between categories of generalized enriched multicategories internal multicategories, provided satisfies suitable property, which holds for several examples. We verify, moreover, that left adjoint is fully faithful, preserves pullbacks, copower functor {Set} <mml:mi>Set</mml:mi> faithful. also apply this result to study descent theory multicategorical structures. These results are built upon base-change which, in turn, was carried out context horizontal lax algebras arising 2-category pseudodouble categories.