VAN KAMPEN COLIMITS AS BICOLIMITS IN SPAN. Tobias Heindel and Paweł Sobociński CALCO 10/09/09 Udine

Transcription

1 VAN KAMPEN COLIMITS AS BICOLIMITS IN SPAN Tobias Heindel and Paweł Sobociński CALCO 10/09/09 Udine

2 INITIAL OBJECT Let C be a category with pullbacks. initial object: 0

3 INITIAL OBJECT Let C be a category with pullbacks. initial object: 0! X

4 INITIAL OBJECT Let C be a category with pullbacks. initial object: 0! X C Par(C)

5 INITIAL OBJECT Let C be a category with pullbacks. initial object: 0! X n 0 m P Q g X f X C Par(C) m, n mono

6 INITIAL OBJECT Let C be a category with pullbacks. initial object: 0! X n 0 m P Q f X C Par(C) m, n mono g X m! =1 0 m!m =1 0 m = m1 P!m =1 P

7 INITIAL OBJECT Let C be a category with pullbacks. initial object: 0! X C Par(C) n 0 m P Q f X nm 1 m, n mono g X m! =1 0 m!m =1 0 m = m1 P!m =1 P

8 INITIAL OBJECT Let C be a regular category. C Rel(C) Works with extra assumption: 0 X = 0

9 INITIAL OBJECT Let C be a regular category. C Rel(C) n 0 m P Q f X P (m,f) 0 X mono g X Works with extra assumption: 0 X = 0

10 INITIAL OBJECT Let C be a category with pullbacks. C Span(C)

11 INITIAL OBJECT Let C be a category with pullbacks. C Span(C) n 0 m P f X Q g X

12 INITIAL OBJECT Let C be a category with pullbacks. C Span(C) n 0 m P Q f X Need extra assumption: g X 0 is strict.

13 INITIAL OBJECT Let C be a category with pullbacks. C Span(C) n 0 m P Q f X Need extra assumption: g X 0 is strict. In fact, 0 is initial in Span(C) iff it is strict (bi)initial in C!

14 TALK IN ONE SLIDE

15 TALK IN ONE SLIDE other examples of properties of colimits

16 TALK IN ONE SLIDE other examples of properties of colimits sums in extensive cats, pushouts in adhesive cats,...

17 TALK IN ONE SLIDE other examples of properties of colimits sums in extensive cats, pushouts in adhesive cats,... Q. Where do such properties come from?

18 TALK IN ONE SLIDE other examples of properties of colimits sums in extensive cats, pushouts in adhesive cats,... Q. Where do such properties come from? many are instances of general condition for colimits called the Van Kampen property. But what does this really mean?

19 TALK IN ONE SLIDE other examples of properties of colimits sums in extensive cats, pushouts in adhesive cats,... Q. Where do such properties come from? many are instances of general condition for colimits called the Van Kampen property. But what does this really mean?

20 TALK IN ONE SLIDE other examples of properties of colimits sums in extensive cats, pushouts in adhesive cats,... Q. Where do such properties come from? many are instances of general condition for colimits called the Van Kampen property. But what does this really mean? Main Theorem: a colimit is Van Kampen in C iff it is a bicolimit in Span(C) (via canonical embedding).

21 TALK IN ONE SLIDE other examples of properties of colimits sums in extensive cats, pushouts in adhesive cats,... Q. Where do such properties come from? many are instances of general condition for colimits called the Van Kampen property. But what does this really mean? Main Theorem: a colimit is Van Kampen in C iff it is a bicolimit in Span(C) (via canonical embedding).

22 TALK IN ONE SLIDE other examples of properties of colimits sums in extensive cats, pushouts in adhesive cats,... Q. Where do such properties come from? many are instances of general condition for colimits called the Van Kampen property. But what does this really mean? Main Theorem: a colimit is Van Kampen in C iff it is a bicolimit in Span(C) (via canonical embedding). A. Such properties are an axiomatic characterisation (in C) of a universal property (in Span(C))!

23 VAN KAMPEN COLIMITS (Lack & Sobocinski 04, Cockett & Guo) Definition: A colimit diagram κ : F C is Van Kampen when for all functors F : J C, cocones κ : F C and cartesian nat. trans. γ : F F F i γ i F i F u Fj κ j κ i F j F u κ j κ i γ j C C c TFAE κ i. is a colimit diagram ii. are all F ic CF i pullback diagrams

24 EXAMPLE - VK COPRODUCTS A coproduct diagram is VK when, given a commutative diagram: X Z Y TFAE A A + B B i1 i 2 top row is a coproduct diagram two squares are pullbacks

25 OTHER EXAMPLES strict initial objects are VK initial objects pushouts along monos in adhesive categories are VK pushouts A pushout is VK iff given a cube with rear faces pullbackstfae: 1. top face is a pushout 2. front faces are pullbacks omega chains in previous talk... a A A m g D C f c n B m C f d B g n D b

26 BICOLIMITS (Kelly & Street)

27 BICOLIMITS (Kelly & Street) Span(C) is a bicategory: canonical notion of colimits are bicolimit

28 BICOLIMITS (Kelly & Street) Span(C) is a bicategory: canonical notion of colimits are bicolimit

29 BICOLIMITS (Kelly & Street) Span(C) is a bicategory: canonical notion of colimits are bicolimit categories: usually do not talk about equality of objects limits, colimits etc are defined up to isomorphism mediating morphisms are unique

30 BICOLIMITS (Kelly & Street) Span(C) is a bicategory: canonical notion of colimits are bicolimit categories: usually do not talk about equality of objects limits, colimits etc are defined up to isomorphism mediating morphisms are unique

31 BICOLIMITS (Kelly & Street) Span(C) is a bicategory: canonical notion of colimits are bicolimit categories: usually do not talk about equality of objects limits, colimits etc are defined up to isomorphism mediating morphisms are unique 2-categories and (especially) bicategories: usually do not talk about equality of arrows bilimits, bicolimits are defined up to equivalence mediating morphisms are essentially unique

32 MORE CONCRETELY Let J be an ordinary category and M: J B a functor A bicolimit consists of the following data: bic M B pseudo-cocone κ: M bic M M u M i κu M j κ i κ bic M j κ idi =1 κi κ v u =(κ v M u ) κ u

33 UNIVERSAL PROPERTY (existence) for all pseudo cocones λ: M X there exists a pseudo mediating morphism that consists of: an arrow h: bic M X isomorphic 2-cells ϕ i : λ i ( h) κ satisfying: M i λ i X M u M j κ i ϕ i h κu bic M κ j = M i λ i X M u M j λ u λ j h ϕ j κ j bic M

34 UNIVERSAL PROPERTY (essential uniqueness) for any h, h : bic M X, a modification ψ : h κ h κ is ( ξ) κ for a unique 2-cell ξ : h h this implies that any mediating morphisms are essentially unique, ie any two are isomorphic via a unique isomorphism

35 MAIN THEOREM Let C have pullbacks and J-colimits. Let be the canonical inclusion. Then: Γ : C Span(C) κ : F C is Van Kampen in C Γκ iff is a bicolimit in Span(C) Proof sketch: lemmas that allow to pass between C and Span(C) restatement of the universal property of bicolimits so that it matches the VK condition.

36 SOME COROLLARIES C a category with pullbacks: C has a strict initial object iff it has an initial object and it is preserved by the embedding into Span(C) C is extensive iff it has binary sums and these are preserved by the embedding into Span(C) C is adhesive iff it has pushouts along monos and these are preserved by the embedding into Span(C)...

37 INTUITIONS Ordinary universal property of colimits is good enough for C with Γ : C Span(C) we pass into a wilder universe VK colimits are reinforced colimits that are ready for this shock

38 LITTLE EXAMPLE - SYMMETRIES 2!! Not VK in the cat of finite ordinals & fns. Γ 2 ( 1 1 ) ( 1 1 ) id id ( 1 1 ) ( 1 1 ) id id ( 1 1 ) ( 1 1 ) ( tw id 1 ) ( 1 1 ) ( 1 1 ) ( tw id 1 ) ( 1 1 ) ( 1 1 ) ( tw tw 1 ) but only two mediating morphisms, so cannot be a bicolimit VK bicolimits are somehow stable under symmetries

39 FUTURE WORK Characterise the VK colimits in Set or at least the VK pushouts! characterise weakenings of the VK condition by looking at universes between C and Span(C) (like Par(C) or Rel(C)) obtain (useful?) weakenings of adhesive categories etc