WALKER, Charles Robert. Lax Familial Representability and Lax Generic Factorizations. Theory and Applications of Categories. New Brunswick: Mount Allison University, 2020, roč. 35, č. 37, s. 1424-1475. ISSN 1201-561X.
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Základní údaje
Originální název Lax Familial Representability and Lax Generic Factorizations
Autoři WALKER, Charles Robert (36 Austrálie, garant, domácí).
Vydání Theory and Applications of Categories, New Brunswick, Mount Allison University, 2020, 1201-561X.
Další údaje
Originální jazyk angličtina
Typ výsledku Článek v odborném periodiku
Obor 10101 Pure mathematics
Stát vydavatele Kanada
Utajení není předmětem státního či obchodního tajemství
WWW URL
Impakt faktor Impact factor: 0.545
Kód RIV RIV/00216224:14310/20:00117741
Organizační jednotka Přírodovědecká fakulta
UT WoS 000594117700037
Klíčová slova anglicky generic factorizations; lax conical colimit of representables
Štítky rivok
Příznaky Mezinárodní význam, Recenzováno
Změnil Změnila: Mgr. Marie Šípková, DiS., učo 437722. Změněno: 13. 1. 2021 15:34.
Anotace
A classical result due to Diers shows that a copresheaf F: A -> Set on a category A is a coproduct of representables precisely when each connected component of F's category of elements has an initial object. Most often, this condition is imposed on a copresheaf of the form B (X, T-) for a functor T : A -> B, in which case this property says that T admits generic factorizations at X, or equivalently that T is familial at X. A classical result due to Diers shows that a copresheaf F: A -> Set on a category A is a coproduct of representables precisely when each connected component of F's category of elements has an initial object. Most often, this condition is imposed on a copresheaf of the form B (X, T-) for a functor T : A -> B, in which case this property says that T admits generic factorizations at X, or equivalently that T is familial at X. Here we generalize these results to the two-dimensional setting, replacing A with an arbitrary bicategory A, and Set with Cat. In this two-dimensional setting, simply asking that a pseudofunctor F: A -> Cat be a coproduct of representables is often too strong of a condition. Instead, we will only ask that F be a lax conical colimit of representables. This in turn allows for the weaker notion of lax generic factorizations (and lax familial representability) for pseudofunctors of bicategories T : A -> B. We also compare our lax familial pseudofunctors to Weber's familial 2-functors, finding our description is more general (not requiring a terminal object in A), though essentially equivalent when a terminal object does exist. Moreover, our description of lax generics allows for an equivalence between lax generic factorizations and lax familial representability. Finally, we characterize our lax familial pseudofunctors as right lax F-adjoints followed by locally discrete fibrations of bicategories, which in turn yields a simple definition of parametric right adjoint pseudofunctors. We also compare our lax familial pseudofunctors to Weber's familial 2-functors, finding our description is more general (not requiring a terminal object in A), though essentially equivalent when a terminal object does exist. Moreover, our description of lax generics allows for an equivalence between lax generic factorizations and lax familial representability. Finally, we characterize our lax familial pseudofunctors as right lax F-adjoints followed by locally discrete fibrations of bicategories, which in turn yields a simple definition of parametric right adjoint pseudofunctors.
VytisknoutZobrazeno: 3. 10. 2022 19:28