Bicategories of Fractions Revisited Dorette Pronk 1 with Laura Scull - PowerPoint PPT Presentation

Bicategories of Fractions Revisited Dorette Pronk 1 with Laura Scull 2 1 Dalhousie University, Halifax, NS 2 Fort Lewis College, Durango, CO FMCS 2019 University of Calgary, Kananaskis Field Station May 2019

Outline Localization 1 Weaker Bicalculus of Fractions Conditions 2 2-Cell Representatives 3 Application to Orbifolds 4

Localization Localization of a Bicategory Let W be a clas of arrows in a bicategory B . A localization of B with respect to W is given by a pseudofunctor J W : B ! B ( W � 1 ) such that for each w 2 W , J W ( w ) is an internal equivalence in B ( W � 1 ) ; J W is universal: composition with J W gives an equivalence, Ps W ( B , C ) ' Ps ( B ( W � 1 ) , C ) . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 3 / 39

o / o o / Localization (Bi)Categories of Fractions For any class W , a representation of this bicategory can be obtained as follows: Objects are those of B ; 1 Arrows are given by finite zig-zags 2 w 1 f 1 w 2 w n f n ··· where all w i are in W ; 2-Cells are equivalence classes of diagrams that are formal 3 pastings of cells from B such that this would be a valid pasting diagram if all arrows from W were reversed. This bicategory is horribly complicated! D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 4 / 39

O o c ; # { ✏ / Localization With a Calculus of Fractions If W satisfies the conditions to admit a bicalculus of fractions, the bicategory B ( W � 1 ) is given by: Objects are those of B ; 1 Arrows are spans 2 w f with w in W ; 2-Cells are equivalence classes of diagrams 3 w f u α β u 0 w 0 f 0 where wu , w 0 u 0 2 W . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 5 / 39

Localization Better, but not good enough The hom-categories can still have proper classes as objects. The equivalence classes of 2-cell diagrams are a priori a bit mysterious and difficult to work with. D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 6 / 39

Localization Solutions To obtain small hom-categories require that W be locally small : for each object A in B , the collection of arrows in W with codomain A is small. wu A In order to make W locally small, it is desirable to work with a weaker set of conditions to obtain a bicalculus of fractions. To obtain a better handle on the 2-cells, we will consider conditions under which there are canonical representatives . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 7 / 39

Localization Solutions To obtain small hom-categories require that W be locally small : for each object A in B , the collection of arrows in W with codomain A is small. we un A In order to make W locally small, it is desirable to work with a weaker set of conditions to obtain a bicalculus of fractions. To obtain a better handle on the 2-cells, we will consider conditions under which there are canonical representatives . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 7 / 39

Localization Solutions To obtain small hom-categories require that W be locally small : for each object A in B , the collection of arrows in W with codomain A is small. uh we A In order to make W locally small, it is desirable to work with a weaker set of conditions to obtain a bicalculus of fractions. To obtain a better handle on the 2-cells, we will consider conditions under which there are canonical representatives . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 7 / 39

Weaker Bicalculus of Fractions Conditions Conditions WBF1-2 The class W ✓ B 1 admits a bicalculus of fractions if it satisfies: WBF1=BF1 All identities are in W . v w BF2 For B / C / D and v , w 2 W , wv 2 W . v w u WBF2 For B / C / D and v , w 2 W , there is A / B in B such that wvu 2 W . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 8 / 39

Weaker Bicalculus of Fractions Conditions Conditions WBF1-2 The class W ✓ B 1 admits a bicalculus of fractions if it satisfies: WBF1=BF1 All identities are in W . v w BF2 For B / C / D and v , w 2 W , wv 2 W . v w u WBF2 For B / C / D and v , w 2 W , there is A / B in B such that wvu 2 W . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 8 / 39

✏ ✏ ✏ / Weaker Bicalculus of Fractions Conditions Condition WBF3 WBF3=BF3 For each A w / B C f with w 2 W , there is an invertible 2-cell, h D A α v w = ) / B C f with v 2 W . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 9 / 39

6 6 6 ( ( 6 ( Weaker Bicalculus of Fractions Conditions Condition WBF4 WBF4=BF4 For each B f w ( B 0 A with w 2 W α + g w B there exists a lifting A ˜ w f A 0 with ˜ B w 2 W ˜ α + g w ˜ A such that α � ˜ v = w � ˜ α . The collection of pairs ( ˜ w , ˜ α ) needs to be suitably compatible. Unit BF4 as BF4, but require ˜ w = id. Unit Co-BF4 If α : fu ) gu and u 2 W , then there is a 2-cell α : f ) g such that ˜ ˜ α u = α . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 10 / 39

6 6 6 ( ( 6 ( Weaker Bicalculus of Fractions Conditions Condition WBF4 WBF4=BF4 For each B f w ( B 0 A with w 2 W α + g w B there exists a lifting A ˜ w f A 0 with ˜ B w 2 W ˜ α + g w ˜ A such that α � ˜ v = w � ˜ α . The collection of pairs ( ˜ w , ˜ α ) needs to be suitably compatible. Unit BF4 as BF4, but require ˜ w = id. Unit Co-BF4 If α : fu ) gu and u 2 W , then there is a 2-cell α : f ) g such that ˜ ˜ α u = α . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 10 / 39

6 6 6 ( ( 6 ( Weaker Bicalculus of Fractions Conditions Condition WBF4 WBF4=BF4 For each B f w ( B 0 A with w 2 W α + g w B there exists a lifting A ˜ w f A 0 with ˜ B w 2 W ˜ α + g w ˜ A such that α � ˜ v = w � ˜ α . The collection of pairs ( ˜ w , ˜ α ) needs to be suitably compatible. Unit BF4 as BF4, but require ˜ w = id. Unit Co-BF4 If α : fu ) gu and u 2 W , then there is a 2-cell α : f ) g such that ˜ ˜ α u = α . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 10 / 39

Weaker Bicalculus of Fractions Conditions Condition WBF5 WBF5=BF5 When w 2 W and there is an invertible 2-cell α : v ) w , then v 2 W . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 11 / 39

�   � �  �   � �  Weaker Bicalculus of Fractions Conditions Adjusted Horizontal Composition Instead of composing by ¯ ¯ w 2 f 1 γ w 1 f 2 w 2 f 1 where γ is a chosen square as in condition BF3, we now compose by ¯ ¯ w 2 u f 1 u γ u w 1 f 2 w 2 f 1 where w 1 ¯ w 2 u 2 W (with u a chosen arrow as in BF2). D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 12 / 39

. / o $ o / w ' / O ✏ / ✏ / g ✏ / 7 ( ( i ✏ / / / ✏ ✏ ✏ o o Weaker Bicalculus of Fractions Conditions Key Technical Lemma w 2 w 1 f Given arrows with w 1 , w 2 2 W and any two squares, ¯ ˜ f f ¯ w 1 ˜ w 1 w 1 α � w 1 β � w 2 w 2 f f with w 2 ¯ w 1 , w 2 ˜ w 1 2 W , there is a diagram, with ¯ ¯ w 1 f s ε δ t ˜ ˜ w 1 f w 2 ¯ w 1 s , w 2 ˜ w 1 t 2 W , δ , ε invertible 2-cells and ¯ f s ε s ˜ ¯ δ f f t ⌘ ˜ w 2 ¯ w 1 w 1 ✏ β w 1 ✏ α w 1 ¯ f f D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 13 / 39

Weaker Bicalculus of Fractions Conditions Remarks This result enables us to define the associativity isomorphisms. It also enables us to use similar techniques to the one given for horizontal composition of arrows to define vertical composition of 2-cell diagrams and whiskering of 2-cells with arrows. D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 14 / 39

Weaker Bicalculus of Fractions Conditions Associativity Isomorphism In of an i var a se tis he i i 02 associativity isomorphism D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 15 / 39

/ v g 7 ✏ 7 v ( O ( O 6 o h 6 ✏ O h ✏ g O w ' w ✏ ' Weaker Bicalculus of Fractions Conditions Vertical Composition Vertical compositon of 2-cell diagrams u 1 f 1 u 2 f 2 v 3 v 1 + α 1 + β 1 and + α 2 + β 2 v 2 v 4 u 2 u 3 f 2 f 3 . is given by v 1 u 1 f 1 v 2 v 2 3 ¯ v 0 ˜ α 1 u 2 β 1 u 2 f 2 ¯ ¯ ˜ ˜ δ δ 2 ¯ α 2 v 0 ˜ u 2 β 2 v 3 v 3 u 3 f 3 v 4 D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 16 / 39

✏ ) 5 ) i O / : o 5 ✏ O ) O 5 5 ) O u / o % Weaker Bicalculus of Fractions Conditions Left Whiskering To calculate u 1 f 1 s 1 g v α β s 2 u 2 f 2 we construct ¯ f 1 ¯ v 1 s 0 1 ˜ u 1 δ 1 f 1 ˜ s 1 γ 1 w 1 t 1 v 1 g v δ 3 β s 2 γ 2 w 2 t 2 ✏ ˜ v 2 f 2 δ 2 s 0 2 ˜ u 2 ✏ ¯ v 2 ¯ f 2 w 2 and lift with respect to v . D. Pronk, L. Scull (Dalhousie, Fort Lewis) Bicategories of Fractions Revisited May, 2019 17 / 39