Arrow-theoretic differential theory

Transcription

1 Arrow-theoretic dierential theory Urs Schreiber August 17, 2007 Abstract We propose and study a notion o a tangent (n + 1)-bundle to an arbitrary n-category Despite its simplicity, this notion turns out to be useul, as we shall indicate Contents 1 Introduction 2 2 Main results 3 21 Tangent (n + 1)-bundle 3 22 G-Flows on Categories Vector ields and Lie derivatives Inner automorphism n-groups Supercategories Categorical lows and quotients 8 23 Curvature and Bianchi Identity or unctors General unctors Curvature and inner automorphisms Parallel transport unctors and dierential orms Sections and covariant derivatives 11 3 Dierential arrow theory Tangent categories The tangent (n + 1)-bundle as a projection The tangent (n + 1)-bundle as a iber-assigning unctor Dierentials o unctors Sections o unctors 20 4 Parallel transport unctors and their curvature Principal parallel transport Trivial G-bundles with connection 20 5 The last section 22 1

2 1 Introduction Various applications o (n-)categories in quantum ield theory indicate that (n- )categories play an important role over and above their more traditional role as mere organizing principles o the mathematical structures used to describe the world: they appear instead themselves as the very models o this world For instance there are various indications that thinking o coniguration spaces and o physical processes taking place in these as categories, with the conigurations orming the objects and the processes the morphisms, is a step o considerably deeper relevance than the tautological construction it arises rom seems to indicate While evidence or this is visible or the attentive eye in various modern mathematical approaches to aspects o quantum ield theory or instance [1], [2] but also [3] the development o this observation is clearly impeded by the lack o understanding o its ormal underpinnings I we ought to think o coniguration spaces as categories, what does that imply or our ormulation o physics involving these coniguration spaces? In particular: how do the morphisms, which we introduce when reining traditional spaces rom 0-categories to 1-categories, relate to existing concepts that must surely secretly encode the inormation contained in these morphisms Like tangent spaces or instance Possibly one o the irst places where this question was at all realized as such is Isham s [4] That this is a piece o work which certainly most physicists currently won t recognize as physics, while mathematicians might not recognize it as interesting mathematics, we take as urther indication or the need o a reined ormal analysis o the problem at hand Several o the things we shall have to say here may be regarded as an attempt to strictly think the approach indicated in Isham s work to its end Our particular goal here is to indicate how we may indeed naturally, generally and useully relate morphisms in a category to the wider concept o tangency For instance his arrow ields on categories we identiy as categorical tangents to identity unctors on categories and ind their relation to ordinary vector ields as well as to Lie derivatives, thereby, by the nature o arrow-theory, generalizing the latter concepts to essentially arbitrary categorical contexts While there is, or reasons mentioned, no real body o literature yet, which we could point the reader to, on the concrete question we are aiming at, the reader can ind inormation on the way o thinking involved here most notably in the work o John Baez, the spiritus rector o the idea o extracting the appearance o n-categories as the right model or the notion o state and process in physics In particular the text [7] as well as the lecture notes [8] should serve as good background reading On the other hand, there are indications that possibly large parts o the n- categorical picture o n-dimensional quantum ield theory which we have in mind already secretly exist in disguise: some o the most sophisticated approached to quantum ield theory known, like BV-quantization methods, heavily rely on L -algebras as well as, dually, quasi-ree dierential graded algebras But, as 2

3 discussed or instance in [5], L -algebras concentrated in the irst n degrees are canonically equivalent to n-categorical analogues o ordinary Lie algebras called Lie n-algebras While equivalent, in many cases realizing the n-categorical structure behind n-term L -algebras is necessary to obtain a coherent conceptual picture o what is really going on Here we shall demonstrate this by describing the theory o n-connections and their (n + 1)-curvature in two parallel ways: once in the integral picture o tangent n-groupoid and once in the corresponding dierential picture [6] o Lie n-algebra and Lie n-algebroids The work that our particular developments here have grown out is described in [9] Our discussion o the Bianchi identity or n-unctors should be compared with the similar but dierent constructions in the world o n-old categories given in [10] and in the context o posets in [16] I thank Bruce Bartlett, David Roberts, Jim Stashe, Sean Tilson or general discussion o the notion o tangent categories described here Special thanks go to Roberto Conti or pointing out the work by Roberts and Ruzzi to me, part o which we shall reproduce below, and to Calin Lazaroiu or pointing out his work [17] to me, which happened to have secretly some overlap with other parts o our discussion here 2 Main results Our working model or all concrete computations in the ollowing is 2Cat, the Gray category whose objects are strict 2-categories, whose morphisms are strict 2-unctors, whose 2-morphisms are pseudonatural transormations and whose 3- morphisms are modiications o these It is clear that all our statements ought to have analogs or weaker, more general and higher n versions o n-categories But with a good general theory o higher n-categories still being somewhat elusive, we won t bother to try to go beyond our model 2Cat So we shall now set n = 2 once and or all and take the liberty o using n instead o 2 in our statements, to make them look more suggestive o the general picture which ought to exist 21 Tangent (n + 1)-bundle We deine or any n-category C an n-category T C which is an (n + 1)-bundle p : T C Obj(C) over the space o objects o C This we address as the tangent bundle o C The deinition o this tangent bundle is morally similar to but in detail somewhat dierent orm the way tangent bundles are deined in synthetic dierential geometry and in supergeometry: we consider the category pt := { } 3

4 as an arrow-theoretic model or the ininitesimal interval or the superpoint in that it is a pued-up version o the mere point pt := { } to which it is equivalent, by way o the injection pt pt, but not isomorphic This suble dierence, rooted deeply in the very notion o category theory, we claim useully models the notion o tangency as extension which hardly diers rom no extension Concretely, we consider T C Hom ncat (pt, C) to be that subcategory o morphisms rom the at point into C which collapses to a 0-category ater pulled back to the point pt The characteristic property o the tangent (n+1)-bundle is that it sits inside the short exact sequence Mor(C) T C C For later use notice that dual to its realization as a projection T C Obj(C) the tangent bundle may be thought o as an n-unctor T C : C op ncat which sends objects a to the tangent categories T a C over them and sends morphisms the the pullback o these along them T C : ( a ) b ( T a C T C T b C ) 22 G-Flows on Categories It is important that the notion o tangent category does not necessarily involve the concept o the ininitesimal We ind that one strength o the notion o tangent categories is that it captures ordinary vector ields on maniolds just as well as exotic generalizations o these, like tangent stacks or odd vector ields as they appear in supergeometry The crucial property o the n-category o sections Γ(T C) o the tangent n-category is that it inherits a monoidal structure, in act the structure o an (n + 1)-group, through a canonical inlcusion Γ(T C) T IdC (Aut(C)) 4

5 We speak o the right hand here as the inner automorphism (n + 1) group INN(C) := T IdC (Aut(C)) o C In this context the image o the above inclusion is denoted Γ(T C) := INN 0 (C) We ind that dierent notions o vector ields and their generalizations correspond to dierent kinds o group homomorphisms v : G Γ(T C) o this group o sections o the tangent category Ordinary vector ields correspond to smooth R-lows on path groupoids Tangent stacks o orbiolds correspond to R-lows o the corresponding Lie groupoids Images o Z 2 and other abelian groups in the category o sections o the tangent category appear in the context o supergeometry (playing the role o certain odd vector ields ) 221 Vector ields and Lie derivatives Let X be a smooth maniold and let P 1 (X) be the groupoid o thin homotopy classes o paths in X Then ordinary vector ields v Γ(T X) on X are in canonical bijection with smooth 1-parameter amilies o categorical tangent vectors to the identity map on P 1 (X): { } Γ(T X) R T (Aut(P IdP1 (X) 1(X))) Relation to Chris Isham s work On a general category C, it may be useul to consider generalizations o this where R is replaced by some other group G The arrow ields on a category C, considered by Isham in [4], are Z-lows on C {Z T IdC (End(C))} Lie derivatives There is a dierential analog o all the structures we are discussing here, with Lie n-groupoids replaced by Lie n-algebroids Possibly a helpul way, in this context, to appreciate the above conception o a vector ield as a smooth group homomorphism Id t P 1 (X) exp(v)(t) Ad exp(v)(t) P 1 (X) 5

6 Id integrally t P 1 (X) exp(v)(t) P 1 (X) exp(v) : R INN(P 1 (X)) Ad exp(v)(t) 0 dierentially Ω (X) ι v Ω (X) d exp(v) inn(ω (X)) L v=[d,ι v] Figure 1: That amilies o categorical inner automorphisms can useully be thought o as vector ields is made particularly plausible by noticing that, in the dierential picture corresponding to a vector ield low by inner automorphisms on the path groupoid, the Lie derivative on dierential orms is a derivation connected, by Cartan s amous ormula, to the 0-derivation is to notice that an ordinary Lie derivative on dierential orms is itsel a derivation o the graded-commutative algebra o dierential orms which is, by Cartan s amous ormula, connected by a chain homotopoy (given by contraction with the vector ield) to the 0-derivation: 0 Ω (X) ι v Ω (X) L v=[d,ι v] More on that in [6] 222 Inner automorphism n-groups O particular importance are the tangent bundles, in our sense, to n-categories which are 1-object (n 1)-groupoids ΣG (n), hence n-groups G (n) In our context these (n 1)-groupoids must be thought o as 1-point orbiolds Accordingly, they have just a single tangent space (tangent n-category) T ΣG (n) This turns out to have interesting properties [15]: For G an ordinary group, one has that T ΣG T IdΣG (Aut(ΣG)) 6

7 is a 2-group, which we call INN(G) It sits inside the exact sequence Z(G) INN(G) AUT(G) OUT(G) o 1-groupoids Here Z(G) is the categorical center o ΣG (which coincides with the ordinary center o G), regarded as a 1-object groupoid This identiies INN(G) as the 2-group o inner automorphisms o G But INN(G) also sits inside the exact sequence G INN(G) ΣG Moreover, it is equivalent to the trivial 2-group, hence contractible This identiies INN(G) as the categorical version o the universal G-bundle For G (2) a strict 2-group, one inds that T IdΣG(2) (Aut(ΣG (2) )) is a 3-group, which we call INN(G (2) ) It sits inside the exact sequence Z(G (2) ) INN(G (2) ) AUT(G (2) ) OUT(G (2) ) o 2-groupoids Here Z(G (2) ) is the 2-categorical center o ΣG, regarded as a 1-object 2-groupoid Inside INN(G (2) ) we have INN 0 (G (2) ), the image o the inclusion This sits inside the exact sequence T ΣG (2) T IdΣG(2) (2Cat) G (2) INN 0 (G (2) ) ΣG (2) Moreover, it is equivalent to the trivial 3-group, hence contractible This identiies INN 0 (G (2) ) as the categorical version o the universal G (2) - 2-bundle 223 Supercategories In the context o supergeometry one encounters categories which exhibit actions by certain abelian groups, usually extensions G Z 2 o Z 2 But oten more is true: there are speciied graded isomorphisms relating any object with its shited copies The basic example or this is the category Vect[Z 2 ] s o ordinary super-vector spaces ie the symmetric braided monoidal category o Z 2 -graded vector spaces equipped with the unique nontrivial symmetric braiding: Z 2 acts on this category by changing the degree o all vector spaces V sv The existence o the canonical isomorphisms s V : V sv distinguishes Vect[Z 2 ] s rom the generic category with a Z 2 -action 7

8 In our arrow-theoretic dierential language, we can express this by the act that on Vect[Z 2 ] s there is an odd vector ield in that we have a Z 2 -low s : Z 2 Γ(T Vect[Z 2 ] s ) One inds that the 2-category o categories with G-low is canonically isomorphic to the 2-category o categories with G-shits deined in [17] and ound useul there or the discussion o categories o graded D-branes 224 Categorical lows and quotients The existence o a G-low on a category C is essentially distinguished by that o a mere G-action by the act that there exists morphisms in C (the G-low lines ) which connect the objects that are related by the G-action For C any 1-category and ρ : G Aut(C) an action o the group G on C by automorphisms, the weak coequalizer o this action C Id ρ(g) C C/G is the category generated rom C and rom a collection o new morphisms a s ag ρ(g)(a) or all a Obj(C) and all g G, subject to the relations a b s a(g) s b (g) ρ(g)(a) ρ(g)() ρ(g)(b) and ρ(g)(a) s a(g) g a ρ(g g)(a) s a(g g) or all g, g G and all Mor(C) This is essentially the descent through the G-action as discussed in [11] 8

9 The quotient category C/G obtained this way is special in that, while it still canonically admits a G-action, now this G-action is inner, in that we have a G-low G INN(C/G) In [17] C/G is called the skew category associated with the category C with G action 23 Curvature and Bianchi Identity or unctors We give a purely arrow-theoretic deinition o the dierential o an arbitrary n-unctor, show that this notion satisies some o the general properties one would wish such a dierential to satisy and demonstrate that in the suitable special case it does indeed reproduce the exterior derivative on dierential orms, including its nonabelian generalizations 231 General unctors Using the unctorial incarnation T C : C op ncat o the tangent bundle, we may push orward any n-unctor F : C D to a connection on the tangent bundle o C, simply by postcomposing δf : C F D T D ncat The crucial point o this construction is that it extends uniquely (up to equivalence) to an (n + 1)-unctor on the (n + 1)-category δf : C (n+1) ncat C (n+1) := Codisc(C) which is obtained rom C by replacing all Hom-(n 1)-categories by the corresponding codiscrete n-groupoids over them By introducing the terminology δf is the curvature o F F is lat i δf is degenerate (sends all (n + 1)-morphisms to identities) we obtain the technically easy but conceptually important generalization o the Bianchi identity: or any unctor F δf is lat or equivalently δδf is degenerate 9

10 232 Curvature and inner automorphisms By combining our deintion o curvature o an n-unctor using the arrow-theoretic dierential with the discussion in 222 we arrive at the statement that the curvature o an n-unctor tra : C ΣG (n) taking values in an n-group G (n) is an (n + 1)-unctor curv = δtra : C ΣINN 0 (G (n) ) taking values in the inner automorphism (n + 1)-group o G (n) Accordingly, the curvature o the curvature, hence the Bianchi identity, takes values in INN 0 INN 0 (G (n) ) Comparison with Roberts-Ruzzi For n = 1 almost this statement had been proposed in [16] as the right ramework or discussing curvature and Bianchi identities The 1-, 2- and 3-groupoids 1G, 2G and 3G which they use are the quotients o our inner automorphism (n + 1)-roups by the respective categorical centers 1G = ΣG (1) 2G = Σ(INN(G)/Z(G)) (2) 3G = Σ(INN 0 (2G)/Z(2G)) (3) We shall indicate in 24 that by orming these quotients here one loses some crucial properties We now describe how our abstract arrow-theoretic notion o curvature does reproduce the amiliar one on dierential orms in suitable smooth context 233 Parallel transport unctors and dierential orms When F : C D is the smooth parallel transport unctor [11] in an n-bundle with connection [12, 13, 14], the arrow-theoretic notion o curvature described above does reproduce the theory o curvature orms o connection orms The general Bianchi identity we have discussed then reduces to the ordinary Bianchi identity amiliar rom dierential geometry More precisely, let C := P 3 (X) be the strict 3-groupoid o thin homotopy classes o k-paths in a smooth maniold X And let G (2) be a strict Lie 2-group coming rom the Lie crossed module H t G α Aut(H) Then, according to [13, 14, 15, 18] we have the ollowing bijections o smooth n-unctors with dierential orms {smooth 1-unctors P 1 (X) ΣG } { A Ω 1 (X, Lie(G)) } { smooth 2-unctors P 2 (X) ΣG (2) } { (A, B) Ω 1 (X, Lie(G)) Ω 2 (X, Lie(H)) F A + t B = 0 } 10

11 (A, B, C) { } smooth 3-unctors P 3 (X) ΣINN(G (2)) C = d A B Now let tra : P 1 (X) ΣG be a smooth 1-unctor with values in the Lie group G Then, under these bijections, we ind that its curvatures correspond to the ollowing dierential orms at top level Ω 1 (X, Lie(G)) Ω 2 (X, Lie(H)) Ω 3 (X, Lie(H)) tra A curv = δtra F A := da + A A δcurv = δδtra d A F A = 0 This way the ordinary Bianchi identity or the curvature 2-orm F A o A is reproduced Notice that or this result come out the way it does, just by turning our abstract crank or dierential arrow-theory, the result o 222 is crucial, which says that the curvature (n + 1)-unctor o a G (n) -transport is itsel an INN(G (n) )-transport 24 Sections and covariant derivatives The curvature curv = dtra o a parallel transport n-unctor is typically trivializable, in that it admits morphisms e : I curv or I some trivial (n + 1)-transport As with the inner automorphism (n + 1)- groups INN(G (n) ), this trivializability, ar rom making these objects uninteresting, turns out to control the entire theory (Compare this to the contractibility o the universal G-bundle: while equivalent to a point, it is ar rom being an uninteresting object, due to the morphisms which go into and out o it According to 222, this comparison is ar more than an mere analogy) A basic act o n-category theory has major implications here: recall that or F and G n-unctors, a transormation G F is given in components itsel by an (n 1)-unctor Now i G is trivial in some sense to be made precise, and i the transormation is an equivalence I F, then this implies that the n-unctor F is entirely encoded in the (n 1)-unctor 11

12 We show that or F = curv = δtra the curvature (n + 1)-unctor o a transport n-unctor tra, the latter essentially encodes the component map o the transormation tra I curv In components this is nothing but a generalization o Stokes law dω = ω X Moreover, it turns out that there may be other trivializations o curv, not by isomorphisms but by mere equivalences On objects, the component unctions o these correspond to sections o the original bundle On morphisms it corresponds, under the identiication o smooth unctors and dierential orms mentioned in??, to the covariant derivative o these sections In [9] it is indicated how all these statements have a quantum analogue as we push our n-unctors orward There it is indicated how the act that transport n- unctors have sections which are themselves transport (n 1)-unctors translates in the context o extended unctorial quantum ield theory to essentially what is known in physics as the holographic principle This needs to be discussed elsewhere, clearly X 12

13 3 Dierential arrow theory 31 Tangent categories Deinition 1 (the point) The point is the n-category pt := { } with a single object and no nontrivial morphisms We shall careully distinguish this rom the n-category pt := { }, consisting o two objects connected by a 1-isomorphism The category pt might be called the at point It is o course equivalent to the point but not isomorphic We ix one injection i : pt pt once and or all It is useul to think o morphisms i : : pt C rom the at point to some codomain C as labeled by the corresponding image o the ordinary point pt C = pt C 311 The tangent (n + 1)-bundle as a projection Deinition 2 (tangent (n + 1)-bundle) Given any n-category C, we deine its tangent (n + 1)-bundle T C Hom ncat (pt, C) to be that sub n-category o morphisms rom the at point into C which collapses to a 0-category when pulled back along the ixed inclusion i : pt pt : the 13

14 morphisms h in T C are all those or which pt pt = pt h C pt C The tangent (n + 1)-bundle is a disjoint union T C = T x C x Obj(C) o tangent n-categories at each object x o C In this way it is an (n + 1)-bundle p : T C Disc(C) over the space o objects o C Example (slice categories) For C any 1-groupoid, ie a strict 2-groupoid with only identity 2-morphisms, its tangent 1-category is the comma category T C = ((Disc(C) C) Id C ) This is the disjoint union o all co-over categories on all objects o C 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 b in C 14

15 Example (strict tangent 2-groupoids) The example which we are mainly interested in is that where C is a strict 2-groupoid For a any object in C, an object o T a C is a morphism q a b A 1-morphism in T a C is a illed triangle q b a F q b in C Finally, a 2-morphism in T a C looks like q b a F F L q b The composition o these 2-morphisms is the obvious one Proposition 1 The tangent (n + 1)-bundle o C its into the short exact sequence Mor(C) T C C Proposition 2 The tangent (n + 1)-bundle T C is a deormation retract o the underlyin space o objects in that it is equivalent to the discrete n-category over the set o objects o C T C Obj(C) This equivalence goes through also i everything is internal to smooth spaces Proo This is to say that each tangent n-groupoid T a C, or all a Obj(C), is equivalent to the trivial n-category T a C pt To establish this, it is suicient to exhibit an invertible transormation rom the identity on T a C to the unctor which sends everything to the identity morphism on a Now making explicit use o our assumption that we are working with strict 2-categories, we can take the components o this on objects a b 15

16 to be simply the morphism ( a b ) a b Id a 1 There is then a unique choice or the component o the transormation on morphisms Remark This result shows in which sense one should think o our tangent n-categories: the right intuition is to think o the space o objects o C as the space in question, and o the morphisms and higher morphisms o C as encoding tangency relations among these objects This means that the way we are to think here o (n-)categories as spaces is the way in which in the theory o stacks orbiolds are regarded as groupoids, rather than, say, the identiication o categories with spaces induced by the nerve construction 312 The tangent (n + 1)-bundle as a iber-assigning unctor The tangent (n + 1)-bundle o a category comes equipped with a canonical parallel transport over the opposite o the category: Deinition 3 For any n-category C, let T C : C op ncat be the n-unctor which sends x T x C or each x Obj(C) and which sends a morphism x T C : T y C T x C y to the n-unctor given by y h z h z r x y h z h z r 16

17 Proposition 3 This unctor respects the morphism T C C rom proposition 1 in that commutes or all morphisms in C T C T x C T y C C This way we even get an n-unctor T : Cat TCat (ncat) as ollows Deinition 4 Deine an n-unctor T : Cat T op Cat (ncat) as ollows It sends any n-category C to the morphism It sends any n-unctor to the triangle C op F D op T C : C op ncat F : C D T C T D ncat with the only obvious transormation illing this triangle The act mentioned in proposition 2, that T C Obj(C) reads in terms o the unctor T C : C op Cat as ollows Proposition 4 For any n-groupoid C, let pt : C op ncat be the n-unctor which sends everything to the identity on the n-category pt Then there is an equivalence pt, C op ncat T C 17

18 32 Dierentials o unctors For any n-unctor with D an n-groupoid, let C F D C (n+1) := Codisc(C) be the (n + 1)-category whose Hom-categories are the codiscrete n-groupoids over the Hom-(n 1)-categories o C (ie with a unique n-morphism or every pair o parallel (n 1)-morphisms) and set C op (δf ) n : F ngrpd D op T D Proposition 5 (δf ) n extends essentially uniquely to an (n + 1)-unctor δf : C (n+1) ngrpd Proo As a warmup, to see the idea, consider n = 1 Then or any two parallel morphisms γ x γ y in C we need a transormation T F (γ) D T F (x) (D) T F (y) (D) T F (γ ) D By writing out what this must be like, and using the act that D is assumed to be an n-groupoid, one inds that there is, up to equivalence, only one possible choice The case n = 2 is entirely analogous 18

19 Notice that δ commutes with pullbacks: given an n-unctor p : C C and a n-unctor F : C D we write p F : C p C F D and call this the pullback o F along p Then we trivially have the ollowing important statement: Proposition 6 The operation δ commutes with pullbacks Hence or all p and all F p (δf ) n = δ(p F ) n Deinition 5 (curvature) We say that δf is the curvature o F An n-unctor is degenerate i it sends all n-morphisms to identity n- morphisms An n-unctor F is lat i its curvature (n + 1)-unctor δf is degenerate Proposition 7 (Bianchi identity) Let F : C D be a 1-unctor with values in a groupoid D Then δf is lat or equivalently δδf is degenerate Deinition 6 (Stokes theorem) The curvature (n + 1)-unctor is canonically trivialized by the unctor it comes rom, by combination o the transormations rom deinition 4 and proposition 4: C op pt F df ncat := C op F D op pt T C ncat T D 19

20 33 Sections o unctors In our context, we may think o an n-unctor F : C D as encoding [11, 13, 14] an n-bundle with connection on Obj(C) From this point o view one is interested in determining the space o n-sections o this n-bundle, as well as the covariant derivative o these sections with respect to the given n-connection Deinition 7 (sections) A section o F is a morphism into δf Hence a section o an n-unctor is a pair, consisting o an n + 1-unctor F : C D C op together with a transormation E ngrpd (δf ) n : C op F D op E ngrpd e T D 4 Parallel transport unctors and their curvature 41 Principal parallel transport 411 Trivial G-bundles with connection Let S be a category playing the role o the path groupoid o some space For us, a trivial G-bundle with connection on S is a unctor tra : S ΣG which we interpret as sending each morphism in S to the parallel G-transport along it The dierential o this unctor we may think o as the curvature o the parallel transport curv = dtra 20

21 in that the component map o the transormation which it assigns to a 2- morphism Σ is essentially the parallel transport tra( Σ) around the boundary o that 2-morphism Notice that curv sends every point to a iber o the orm T ΣG = INN(G) over it Moreover, by proposition 3 this assignment respects the morphism INN(G) ΣG curv(γ) INN(G) INN(G) ΣG It turns out that curv is trivializable in various ways There is interesting inormation in the morphisms that trivialize it To see this, consider two special bundles with connection over S, the trivial point bundle J : S { } and the trivial G-bundle with trivial connection I : S { } ΣG Proposition 8 There is an isomorphism (not just an equivalence) S 2 di curv whose component map is essentially a map Mor(S) G and as such coincdes with tra Hence we write, with conventient and suggestive abuse o notation, di Gauge equivalent transport unctors tra curv g : tra tra correspond to dierent but equivalent trivializations o the same curvature 2- unctor tra Cat di g curv tra 21

22 There may also be trivialization o curv which correspond to sections o the original bundle These come rom equivalences o curv with the trivial point bundle J S 2 Their component maps pick out an element o G over each object o S e δj curv Cat 5 The last section in which I admit that I am running out o steam or the moment Reerences [1] D S Freed, F Quinn, Chern-Simons Theory with Finite Gauge Group [2] D S Freed Higher Algebraic Structures and Quantization, available as hep-th/ [3] S Willerton, The twisted Drineld double o a inite group via gerbes and inite groupoids, available as math/ [4] C J Isham, A new approach to quantising space-time: I Quantising on a general category, available as arxiv:gr-qc/ [5] J Baez, A Crans, Lie 2-Algebras [6] U S, J Stashe, Structure o Lie n-algebras [7] J Baez, A Lauda, A History o n-categorical Physics [8] J Baez Quantization and Cohomology, lecture notes [9] U S On 2d QFT: From Arrows to Disks [10] A Kock Ininitesimal cubical structure, and higher connections, available as arxiv: [11] U S, K Waldor, Parallel Transport and Functors, available as arxiv: [12] T Bartels, 2-Bundles, available as arxiv:math/ [13] J Baez, U S Higher Gauge Theory, available as arxiv:math/ [14], U S, K Waldor, Parallel 2-Transport and 2-Functors (in preparation) 22

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