Lax Familial Representability and Lax Generic Factorizations
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F20%3A00117741" target="_blank" >RIV/00216224:14310/20:00117741 - isvavai.cz</a>
Result on the web
<a href="http://www.tac.mta.ca/tac/volumes/35/37/35-37.pdf" target="_blank" >http://www.tac.mta.ca/tac/volumes/35/37/35-37.pdf</a>
DOI - Digital Object Identifier
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Alternative languages
Result language
angličtina
Original language name
Lax Familial Representability and Lax Generic Factorizations
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Theory and Applications of Categories
ISSN
1201-561X
e-ISSN
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Volume of the periodical
35
Issue of the periodical within the volume
37
Country of publishing house
CA - CANADA
Number of pages
52
Pages from-to
1424-1475
UT code for WoS article
000594117700037
EID of the result in the Scopus database
2-s2.0-85089849293