Rui Prezado: Change-of-base for generalized multicategories

30th of October, 2024 ——— A multicategory is a categorical structure that consists of objects and multimorphisms*, whose domain consists of a finite (possibly empty) string of objects, and whose codomain consists of a single object. Their composition and identity, as well as the unity and associativity properties, are well modeled by the extension of the *free monoid monad on Set to the (proarrow) equipment Span(Set) of spans in Set. Multicategories, together with their respective functors, can be obtained by considering a suitable kind of algebra with respect to the extended free monoid monad on Span(Set). This abstraction can be carried out with any cartesian monad T on a category A with pullbacks. We have an extension of T to the equipment Span(A) of spans in A, and the respective "T-algebras" are the so-called *T-categories internal to A*, whose category we denote by Cat(T,A). Most importantly, if we are provided with another cartesian monad S on a category B, and a suitable monad morphism (F,\phi) from (A, T) to (B, S)$, it was shown in [3] that (F,\phi) induces a change-of-base functor from Cat(T,A) to Cat(S,B). The enriched counterpart of such generalized multicategories were first considered in [1]; in essence, enriched (T,V)-categories can be obtained as the "T-algebras" for a suitable monad T on the equipment V-Mat, where the enriching category V is a suitable monoidal category. Likewise, these also have a notion of change-of-base functors. In general, we can consider horizontal lax T-algebras [2] for a lax monad T on a pseudodouble category D, special cases of which are enriched and internal generalized multicategories. This talk aims to present notions of change-of-base functors between such horizontal lax algebras, the study of which was motivated by understanding the relationship between enriched and internal multicategorical structures, the main topic of study of [4], joint work with F. Lucatelli Nunes. We assume the basics of double category theory, and some familiarity with multicategories will prove to be worthwhile. 1. M. M. Clementino, W. Tholen. Metric, topology and multicategory -- a common approach. *J. Pure Appl. Algebra*, (179):13--47, 2003. 2. G. Cruttwell, M. Shulman. A unified framework for generalized multicategories. *Theory Appl. Categ.*, 24(21):580--655, 2010. 3. T. Leinster. *Higher Operads, Higher Categories*, volume 298 of London Mathematical Society Lecture Note Series. Cambridge University Press, 2004. 4. R. Prezado, F. Lucatelli Nunes. Generalized multicategories: change-of-base, embedding and descent. To appear in Appl. Categ. Structures.