WALKER, Charles Robert. Lax Familial Representability and Lax Generic Factorizations. Theory and Applications of Categories. New Brunswick: Mount Allison University, vol. 35, No 37, p. 1424-1475. ISSN 1201-561X. 2020.
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Basic information
Original name Lax Familial Representability and Lax Generic Factorizations
Authors WALKER, Charles Robert (36 Australia, guarantor, belonging to the institution).
Edition Theory and Applications of Categories, New Brunswick, Mount Allison University, 2020, 1201-561X.
Other information
Original language English
Type of outcome Article in a journal
Field of Study 10101 Pure mathematics
Country of publisher Canada
Confidentiality degree is not subject to a state or trade secret
WWW URL
Impact factor Impact factor: 0.545
RIV identification code RIV/00216224:14310/20:00117741
Organization unit Faculty of Science
UT WoS 000594117700037
Keywords in English generic factorizations; lax conical colimit of representables
Tags rivok
Tags International impact, Reviewed
Changed by Changed by: Mgr. Marie Šípková, DiS., učo 437722. Changed: 13/1/2021 15:34.
Abstract
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.
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