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SUMMARY:Rui Prezado (Universidade de Aveiro)
DTSTART:20241030T160000Z
DTEND:20241030T170000Z
DTSTAMP:20260409T130927Z
UID:SecondVDCW/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/SecondVDCW/9
 /">Change-of-base for generalized multicategories</a>\nby Rui Prezado (Uni
 versidade de Aveiro) as part of Second Virtual Workshop on Double Categori
 es\n\n\nAbstract\nA *multicategory* is a categorical structure that consis
 ts of objects\nand *multimorphisms*\, whose domain consists of a finite (p
 ossibly empty)\nstring of objects\, and whose codomain consists of a singl
 e object. Their\ncomposition and identity\, as well as the unity and assoc
 iativity\nproperties\, are well modeled by the extension of the *free mono
 id monad*\non $\\mathsf{Set}$ to the (proarrow) equipment\n$\\mathsf{Span}
 (\\mathsf{Set})$ of *spans* in $\\mathsf{Set}$.\nMulticategories\, togethe
 r with their respective functors\, can be\nobtained by considering a suita
 ble kind of algebra with respect to the\nextended free monoid monad on $\\
 mathsf{Span}(\\mathsf{Set})$.\n\nThis abstraction can be carried out with 
 any *cartesian* monad $T$ on a\ncategory $\\mathcal A$ with pullbacks. We 
 have an extension of $T$ to the\nequipment $\\mathsf{Span}(\\mathcal{A})$ 
 of spans in $\\mathcal A$\, and the\nrespective "$T$-algebras" are the so-
 called *$T$-categories internal to\n$\\mathcal A$*\, whose category we den
 ote by $\\mathsf{Cat}(T\,\\mathcal A)$.\nMost importantly\, if we are prov
 ided with another cartesian monad $S$ on\na category $\\mathcal B$\, and a
  suitable monad morphism\n$(F\,\\phi) \\colon (\\mathcal A\, T) \\to (\\ma
 thcal\nB\, S)$\, it was shown in [3] that $(F\,\\phi)$ induces a *change-o
 f-base*\nfunctor $\\mathsf{Cat}(T\,\\mathcal A) \\to\n\\mathsf{Cat}(S\,\\m
 athcal B)$.\n\nThe enriched counterpart of such generalized multicategorie
 s were first\nconsidered in [1]\; in essence\, *enriched $(T\,\\mathcal V)
 $-categories*\ncan be obtained as the "$T$-algebras" for a suitable monad 
 $T$ on the\nequipment $\\mathcal V$-$\\mathsf{Mat}$\, where the enriching 
 category\n$\\mathcal V$ is a suitable monoidal category. Likewise\, these 
 also have\na notion of change-of-base functors.\n\nIn general\, we can con
 sider *horizontal lax $T$-algebras* [2] for a\nlax monad $T$ on a pseudodo
 uble category $\\mathbb D$\, special cases of\nwhich are enriched and inte
 rnal generalized multicategories. This talk\naims to present notions of *c
 hange-of-base functors* between such\nhorizontal lax algebras\, the study 
 of which was motivated by\nunderstanding the relationship between enriched
  and internal\nmulticategorical structures\, the main topic of study of [4
 ]\, joint\nwork with F. Lucatelli Nunes.\n\nWe assume the basics of double
  category theory\, and some familiarity\nwith multicategories will prove t
 o be worthwhile.\n\n1.   M. M. Clementino\, W. Tholen. Metric\, topology a
 nd multicategory -- a\n    common approach. *J. Pure Appl. Algebra*\, (179
 ):13--47\, 2003.\n\n2.   G. Cruttwell\, M. Shulman. A unified framework fo
 r generalized\n    multicategories. *Theory Appl. Categ.*\, 24(21):580--65
 5\, 2010.\n\n3.   T. Leinster. *Higher Operads\, Higher Categories*\, volu
 me 298 of\n    London Mathematical Society Lecture Note Series. Cambridge\
 n    University Press\, 2004.\n\n4.   R. Prezado\, F. Lucatelli Nunes. Gen
 eralized multicategories:\n    change-of-base\, embedding and descent. To 
 appear in *Appl. Categ.\n    Structures.*\n
LOCATION:https://researchseminars.org/talk/SecondVDCW/9/
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