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@article{1727678, author = {Walker, Charles Robert}, article_location = {New Brunswick}, article_number = {37}, keywords = {generic factorizations; lax conical colimit of representables}, language = {eng}, issn = {1201561X}, journal = {Theory and Applications of Categories}, title = {Lax Familial Representability and Lax Generic Factorizations}, url = {http://www.tac.mta.ca/tac/volumes/35/37/3537.pdf}, volume = {35}, year = {2020} }
TY  JOUR ID  1727678 AU  Walker, Charles Robert PY  2020 TI  Lax Familial Representability and Lax Generic Factorizations JF  Theory and Applications of Categories VL  35 IS  37 SP  14241475 EP  14241475 PB  Mount Allison University SN  1201561X KW  generic factorizations KW  lax conical colimit of representables UR  http://www.tac.mta.ca/tac/volumes/35/37/3537.pdf L2  http://www.tac.mta.ca/tac/volumes/35/37/3537.pdf N2  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 twodimensional setting, replacing A with an arbitrary bicategory A, and Set with Cat. In this twodimensional 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 2functors, 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 Fadjoints 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 2functors, 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 Fadjoints followed by locally discrete fibrations of bicategories, which in turn yields a simple definition of parametric right adjoint pseudofunctors. ER 
WALKER, Charles Robert. Lax Familial Representability and Lax Generic Factorizations. \textit{Theory and Applications of Categories}. New Brunswick: Mount Allison University, 2020, roč.~35, č.~37, s.~14241475. ISSN~1201561X.
