Tangent Categories. David M. Roberts, Urs Schreiber and Todd Trimble. September 5, 2007

Transcription

1 Tangent Categories David M Roberts, Urs Schreiber and Todd Trimble September 5, 2007 Abstract For any n-category C we consider the sub-n-category T C C 2 o squares in C with pinned let boundary This resolves the space o objects in C in a natural way We describe various properties o T C and indicate why it deserves to be addressed as the tangent n-category o C Contents 1 Introduction 1 2 Tangent Categories 2 21 Deinition 2 22 Inner automorphisms 4 23 Properties 5 24 Simplicial aspects 6 3 Tangent n-categories 7 31 Strict tangent 2-Categories 7 4 Arrow-ields and Flows on n-categories 7 41 G-lows 8 5 Universal n-bundles Principal 1-Bundles with connection 10 1 Introduction URS: Instead o a real introduction, at the moment I oer only the ollowing relection Tangent categories play two dierent important roles, which a priori seem to be rather unrelated: The tangent n-category T C is a pued up version o the space o objects C 0 For C an n-groupoid, the canonical inclusion C 0 T C 1

2 is an equivalence The canonical sequence Mor(C) T C C is the n-groupoid incarnation o the universal C-bundle [2] At the same time, T C does know about the tangency relations on C 0 induced by Mor(C): or C an n-groupoid, G-lows Γ G (T C) := {G Γ(T C) T IdC (End(C))} do provide a generalization o the concept o vector ields on C 0 in that or G = R and with everything taken to be smooth we have that sections Γ R (T C) Γ(Lie(C)) do coincide with the sections o the Lie n-algebroid associated with C The apparent dichotomy universal C spaces on one hand, dierentials on C on the other is resolved by noticing that T C is actually to be regarded as the universal C-bundle equipped with the universal C-connection [3] 2 Tangent Categories 21 Deinition We write or the terminal category and pt := {} 2 := { } or the category with two objects and one nontrivial morphism, going between them Then or C any category, we have the category C 2 := Hom Cat (2, C) o commuting squares in C, with composition being the vertical pasting o squares C 2 has two obvious projections onto C C 2 dom C cod which may be thought o as arising rom pullback along the two injections pt 2 2

3 in that a 1 a 1 dom : 2 C pt 2 C a 2 Deinition 1 (tangent category) For C any category, its tangent category T C is deined to be the strict pullback T C C 2 a 2 dom in Cat C 0 i C C Here C 0 := Obj(C) is regarded as a discrete category and i C : C 0 C sends objects to identity endomorphisms Hence T C is the co-slice category T C = (a C) a Obj(C) Objects o T C are morphisms : a b in C, and morphisms T C are commuting triangles b h in a h in C b Remark The deinition o the tangent category is an exact analogue o the sum o path spaces, construed as a pullback in Top: P X X I eval 0 X X 3

4 Here I = [0, 1] is the interval, eval 0 is evaluation at the let boundary o the interval and X is the set underlying the topological space X, equipped with the discrete topology That and how T C is still useully thought o as a tangent bundle is discussed in 4 Deinition 2 (tangent category unctor) We write or the corresponding unctor T : Cat Cat Hence or F : C D any unctor, the unctor T F : T C T D acts by postcomposition with F, in that a 1 a 1 T F : 2 C 2 C F D a 2 a 2 22 Inner automorphisms For C any category, the categorical tangent space T IdC (End(C)) in End(C) at the identity endomorphism plays a special role It makes good sense to address these endomorphisms connected to the identity as inner endomorphisms Deinition 3 For C any groupoid, we address as the automorphism 2-group o C; AUT(C) := Aut Cat (C) INN(C) := T IdC (End(C)) as the inner automorphisms 2-group o C as the center o C Z(C) := ΣEnd IdC 4

5 OUT(C) := coker(inn(c) AUT(C)) as the outer automorphism 2-group o C URS: I need to think about how to deine OUT(C) properly These it into an exact sequence o 2-groups 23 Properties Z(C) INN(C) AUT(C) OUT(C) Proposition 1 The discrete category over the space Mor(C) arises as the pullback cod Mor(C) C 0 i C T C C 2 cod C We may read that as a short exact sequence Mor(C) T C C Proposition 2 When C is a groupoid, then T C C 0 and in act the projection T C C 0 is weakly inverse to the canonical section C 0 T C Deinition 4 We write or the category o sections o T C C 0 Γ(T C) e : C 0 T C Proposition 3 When C is a groupoid, then we have a canonical equivalence (isomorphism, even) Γ(T C) T IdC (End(C)) 5

6 Remark Recall rom 22 that this equips Γ(T C) with a monoidal structure, even the structure o a 2-group INN(C) := T Id (End(C)) This monoidal structure is crucial in 4 and or relating tangent categories to ordinary notions o tangent spaces Notice that or n > 1 one inds that Γ(T C) T IdC (End(C)) is no longer an equivalence but becomes a proper inclusion 24 Simplicial aspects Notice that a sequence o k composable morphisms a b c d x in T C is, since all triangles commute, the same as a sequence a b c d x o k + 1 composable morphisms in C This means that when passing to nerves, the tangent unctor T becomes what is known as décalage in the simplicial context: Deinition 5 Denote by [1 + ( )] : op op the obvious unctor which acts on objects as [n] [n + 1] Proposition 4 We have a weakly commuting square Cat T Cat N [ op, Set] N [1+( )] [ op, Set] 6

7 Proo Notice that n-simplices in T C are commuting squares [n] 2 [n] [0] C but the pushout o this co-cone is [n + 1] [n] 2 [n], : where k x 0 x 1 x 2 [0] [n + 1] x k k Hence we unctorially assign (n + 1)-simplices in C to n-simplices in T C x 0 [n] 2 [n] [n + 1]! [0] C a b c d x 3 Tangent n-categories 31 Strict tangent 2-Categories 4 Arrow-ields and Flows on n-categories Isham [1] coined the term arrow ield on a category C or what we conceive as a section e Γ(T C), 7

8 thinking o it as a model or a tangent vector ield on C 0 On the other hand, such an e is ar rom being ininitesimal in any sense We shall now make use o the monoidal structure on Γ(T C) also noticed by Isham to obtain a sensible notion o categorical vector ields We exhibit the special cases in which this reproduces ordinary sections o ordinary vector bundles 41 G-lows Deinition 6 For C any groupoid and G any group, we address a group homomorphism v : G INN(C) as a G-low on C For each g G we write Id C v(g) Ad vc (g) C or the corresponding element in INN(G) A morphism between two categories C and D equipped with G-lows is a unctor which respects the lows in that F : C D Id C v C (g) Ad vc (g) C F = C F Id D D v D (g) D Ad vd (g) or all g G We write Γ G (T C) := Hom(G, INN(C)) or the collection o G-lows on C 8

9 Example (ordinary vector ields) Let X be a smooth maniold and Π 1 (X) its undamental groupoid Smooth R-lows on Π 1 (X) are in canonical bijection with ordinary vector ields on X Γ(T X) Γ R (T Π 1 (X)) (Here on the let T X denotes the ordinary tangent bundle o X) Example (odd vector ields) spaces The parity shit operator Let svect be the category o super vector Π : svect svect characterized by the act that V W ΠV Π ΠW or all morphisms V W with V ΠV the canonical isomorphism maniestly is a Z 2 -low on svect, hence an element Π Γ Z2 (T svect) It might be useul to think o Π as the low o an odd vector ield in supergeometry Example (Lie algebroids) For C any Lie groupoid with Lie algebroid Lie(C), we have Γ R (T C) Γ(Lie(C)) The canonical morphism induces the anchor map C codisc(c 0 ) Γ R (T C) Γ R (T Π 1 (C 0 )) The Lie bracket on sections is obtained rom the group commutator in INN(C) in the usual way 9

10 5 Universal n-bundles 51 Principal 1-Bundles with connection For the ollowing, let C := ΣG be the one-object groupoid given by a group G The sequence Mor(C) T C C which each tangent category sits in then becomes which we also requently denote G T ΣG ΣG, G INN(G) ΣG This sequence is the universal G-bundle in the world o groupoids Proposition 5 (Segal) The geometric realization o the nerve o is a model or the universal G-bundle G T ΣG ΣG G T ΣG ΣG G EG BG But here we shall ind it useul not to pass to spaces by realizing nerves The entire discussion can useully be done entirely within the world o groupoids For X some space, choose a good cover π : Y X and denote by Y [2] π 1 π 2 Y the corresponding groupoid Noticing that Y [2] X we may take this as a groupoid model o X Proposition 6 Equivalence classes o principal G-bundles on X are in bijection with equivalence classes [Y [2], ΣG]/ o unctors 10

11 Proo By unwrapping the relevant deinitions one inds that unctors Y [2] ΣG are precisely G-cocycles on X, while transormations o these unctors are precisely isomorphisms o G-cocycles We may hence regard : Y [2] ΣG as a classiying map By pulling this back along the groupoid version o the universal G-bundle G INN(G) Y [2] ΣG we obtain the groupoid version o the total space Y [2] ΣG INN(G) INN(G) Y [2] ΣG o the G-bundle classiied by But there is more Since or C = ΣG we have that T C is again itsel a 2-group, we may iterate the tangent category construction to obtain Mor(C) T C C T C T ΣT C ΣT C C ΣT C This does not close strictly, but up to pseudonatural transormation Mor(C) T C C T C T ΣT C ΣT C C ΣT C Here the sequence in the middle is the universal INN(G)-2-bundle We may regard 11

12 ΣT C as the undamental 2-groupoid o the universal G-bundle; T ΣT C as the pair groupoid EG EG o the undamental G-bundle, pulled back to the undamental 2-groupoid, such that ater diving out G it becomes the Atiyah groupoid At(EG) := EG G EG pulled back to the undamental 2-groupoid A 2-morphism in ΣT C looks like g h g =hg A 2-morphism in T ΣT C covering this looks like q F F g h g g h g, q where all labels are elements in G Here one should think o q and q as elements in the iber o EG over the chosen base point, and o F and F as two choices o iber isomorphisms over the paths g and g, respectively Since T C = INN(G) is equivalent to the trivial 2-group, its universal 2- bundle T ΣT C is trivializable, and we have a canonical section ΣT C T ΣT C But this section o the universal INN(G) 2-bundle we can regard as a choice o connection on the 1-bundle, namely as a splitting o the Atiyah groupoid projection T ΣT C ΣT C Id g h g g g g h, g Id But this is essentially nothing but the identity 2-unctor ΣT C ΣT C 12

13 We should think o this as the universal connection on the universal G-bundle: curv EG := Id : ΣT C ΣT C Write Π 2 (X) or the undamental 2-groupoid o the space X (objects are the points in X, morphisms are thin-homotopy classes o paths in X, 2-morphisms are homotopy classes o suraces in X) The weak pushout C 2 (Y ) Π 2 (Y [2] ) Π 2 (Y ) Π 1 (Y ) C 2 (Y ), addressed as the path pushout in [4] is the 2-groupoid modelling the undamental 2-groupoid o X with respect to the covering Y It is the 2-groupoid generated rom Π 2 (Y [2] ) and rom Y [2], modulo the relations π 1 (x) π 1(γ) π 1 (y) π 2 (x) π 2(γ) π 2 (y) γ or all x y in Mor 1 (Π 2 (Y [2] )) We ind that extending our classiying map Y [2] C ΣT C rom points to paths Y [2] C 2 (Y ) g C ΣT C (curv,) is the same as choosing a G-connection on the bundle classiied by, by [4] Locally, ie on generators o C 2 (Y ) coming rom Π 2 (Y [2] ) this 2-unctor curv is precisely the curvature 2-unctor o a parallel transport tra : P 1 (Y ) ΣG 13

14 Notice that we may think o the connection on our bundle (curv, ) : C 2 (Y ) ΣT C as the pullback o the universal connection curv EG := Id : ΣT C ΣT C along a reinement o our classiying map, simply as (curv, ) : C 2 (Y ) ΣT C curv EG ΣT C 14

15 Reerences [1] C J, Isham, Quantising on a general category, AdvTheorMathPhys 7 (2003) , available as arxiv:gr-qc/ [2] D M Roberts, U Schreiber, The inner automorphism 3-group o a strict 2-group, available as arxiv: [3] U Schreiber, J Stashe, Connections with values in Lie n-algebras, in preparation [4] U Schreiber, K Waldor, Parallel transport and unctors, available as arxiv:

16 Y [2] Y [2] T C T C C = ΣG C 2 (Y ) (curv,) T ΣT C T ΣT C EG EEG C 2 (Y ) curv EG ΣT C BG BEG (curv,) Lie Ω li (G) i Ω (P ) (A,F A ) g inn(g) Figure 1: The universal G-bundle and its pullbacks in the world o groupoids and Lie algebroids C = ΣG is the one-object groupoid corresponding to G, T C = INN(G) its inner automorphism 2-group, whose underlying groupoid is the total space o the universal G-bundle Y X is a good cover o base space and C 2 (Y ) the undamental 2-groupoid o X relative to this cover Assuming Y = P to be the total space o a G-bundle itsel, dierentiation takes us to the world o Lie algebroids, here presented in terms o their Koszul dual qdgcas, as indicated The geometric realization is indicated only or orientation purposes Notice that INN(G) EG implies that EG has the structure o a topological group 16