Models for synthetic supergeometry

C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES DAVID N. Y ETTER Models for synthetic supergeometry Cahiers de topologie et géométrie différentielle catégoriques, tome 29, no 2 (1988), p. 87-108 <http://www.numdam.org/item?id=CTGDC_1988__29_2_87_0> © Andrée C. Ehresmann et les auteurs, 1988, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET Vol. GÉOMÉTRIE DIFFÉRENTIELLE XXIX - 2 (1988) CATÉGORIQUES MODELS FOR SYNTHETIC SUPERGEOMETRY by David N. YETTER RÉSUMÉ. Synth6tique tielle avec Les notions de base de la G6om6trie Diff6rentielle sont modifi6es pour englober la "geometric diff6renparam6tres commutants et anti-commutants" n6ces- saire pour les theories de supergravit6. Des mod6les analogues aux topos de Dubuc construits, et leurs rapports avec la th6orie et de Stein sont Kostant des de retrouver varietes gradu6es sont explicités. Plusieurs faqons des espaces "bosniques" sont explor6es. Finalernent des resultats 616mentaires de GDS sont étendus au cas supersymétrique : en particulier on montre que la fibre (synth6tique) tangente a 1’unite d’un objet groupe assez r6gulier une structure d’algebre de Lie gradu6e. Llint6gration de superchamp est considérée bri6vement dans le cadre synth6tique. a 0. INTRODUCTION. The enterprise of Synthetic Differential Geometry (SDG), begun Lawvere’s 1967 lecture on "Categorical Dynamics", may be seen as an attempt to axiomatize (hence the name synthetic), and to provide a model theory for the way in which physicists work with smooth phenomena - for example, in SDG vector fields really are infinitesimal flows, or, equivalently, infinitesimal deformations of the identity map, on a manifold. Seen in this light, it is reasonable to attempt to bring the tools of SDG to bear on the construction of mathematical models for supergravity in which a "differential geometry with both commuting and anti-commuting parameteres" is needed. The algebraic-geometric flavor of both Kostant’s theory of graded manifolds and the model theory for SDG as developed by Dubuc and others further suggests the possibility of fruitful interaction. It is the purpose of this paper to begin that work, but only to begin: all results contained herein may be seen either "super" in 87 on SDG [6], or as "synProceedings of the NATO Superspace [9]. We refer readers versions of results contained in Kock’s book thetic" versions of results contained in the Workshop on Mathematical Aspects of unfamiliar with either of the subjects considered to those works as good introductions, and in particular to Kock’s monograph, since many of the proofs of results contained herein are more or less routine generalizations of Kock’s proofs to the "super" case. Proofs of this sort will in general be sketched briefly, as the reader, armed with Kock’s book, can easily fill in the details. Our main result is the construction of "super" analogues of the Dubuc topos and the Stein topos (cf. [4; 51). We go on to consider some properties of these categories, both as models for supergeometry in their own right, and in comparison to Kostant’s theory of graded manifolds [7]J and standard models of SDG (cf. Dubuc [4] and Hoskin [5]). presence of these results, it is hoped that any general sufficiently synthetic proof of a result in differential geometry will carry the "super" result (provided, of course, that some care is taken in how definitions are "superfied" and how integration is handled - the view of Batchelor that Berezin integration is really odd-variable differentiation is undoubtedly correct in the synthetic setting). In 1. the "SUPERFICATION" OF THEORIES. Had the synthetic approach been considered in Batchelor 111, it would have been classed among the "geometric" approaches in that we begin with an algebra with anti-commuting elements. Rather than living in the category of Sets, and being endowed with a topology, this algebra will lie in a Grothendieck topos constructed along with it, and it will be the very structure of the underlying topos that will carry the "geometric" data. The actual construction of that algebra will, however, preliminaries: have to wait until after some algebraic differentially closed theory, T , over K = R or C is equational theory (in the sense of Lawvere [8]), extending the theory of commutative K-algebras, whose n-ary operations are named by infinitely differentiable functions K n -&#x3E; K, and satisfying DEFINITION 1.1. A an 88 Any generalized composition of operations in T names an T, and any equation holding among the functions holds among the operations they name in T. DCT2. If : Kn -&#x3E; K names an operation in T, then so does Kn -&#x3E; K, the I’" partial derivative for I = 1,...,n. DCT1. operation in include the theories of polynomials over R or C (the notion of algebra over the base field), the theory of Coofunctions, and the theory of analytic functions (real or complex). Note that a model of any DCT over K is a fortiori a commutative K-algebra. To introduce anti-commuting elements, we modify the Examples usual theory: superfication S(T) of a DCT T is the theory named by all formal composites of operations in operations T and unary operations, B and F. Two names of operations, and if their values, ||o||v and ||Y||v, in every name the same operation Grassmann Instantiation v are equal. A Grassman instantiation v for an expression in n-variables X1,...,Xn is a choice of a finitely generated Grassmann algebra (over DEFIITII01f 1.2. The whose are K), A, and expression o a vector v = E An. The value ||o||v of an is defined inductively as follows: (V1,...,Vn) in the instantiation v ||X||v = Vi, ||s||v = s for any ||B(o)||v= even and if f: Kn -&#x3E; K is where I ranges part an over K (as a 0-ary operation in T ) , of II; II..., ||F(o)||v = odd part of ||o||v, s e operation of T, then all ordered multiindices i ii ... i. ; r (for all s &#x3E; i), and where and The following theorem makes precise the way in which we have replaced commutativity with (Z /2-)graded commutativity. Throughout the following we use the subscript B to denote the even grade (which we call the bosonic grade) and F denote the odd grade (which we call fer1llionic) . 89 THEOREM 1.3. (2/2-) graded If T is any DCT commuta ti ve K-algebras, K-algebras. commutative commuta ti ve over K- algebra; K, then is the any S (T) -model is a moreover, if T is the theory of then S(T) of theory (Z/2-)graded PROOF. Observe that S(T) has among its operations: for each element s of K, a constant; two binary operations named by multiplication and addition as operations in T; and two unary operations B "bosonic part" and F "fermionic part". Letting it is these easy to show that A is as relevant and B and F grades, equations Grassmann follow instantiation, as from since a graded commutative K-algebra with projections onto the grades: all the the fact that they hold in any Grassmann algebras are themselves graded commutative algebras. Conversely, letting B(x) = degree 0 part of x, and F(x) = degree 1 part of x graded commutative K-algebra becomes an S (K-alg) model. To verify any equation of S (K-alg), it suffices to verify it in the free graded commutative a lgebra over K on n-generators (for n the number of variables in the equation), but this is the Grassmann algebra on 1 = 1,...,n over the F (xi), n-generators algebra polynomial any KIB (xi)|i =1,..., n]. To verify an equation in this algebra, it suffices to verify it sufficiently large finite number of instantiation of the variables B(x¡) by field elements (how many depends on the degree of the equation). But these are simply Grassmann instantiations in the sense above, and the equation must hold in all such. at a The next few propositions Sets) for any DCT T. give wealth a of S(T)-models (in PROPOSITION 1.4. For any K-DCT T, any T -model A becomes an S (T)model when equipped with the operations B(x) = x and F(x) = 0. PROOF. Given any instantiation is equation in S (T), an instantiation algebra of its an part equation even in any Grassmann of T in a Weil over K. (It is easy to show that any Weil algebra - i.e., finite dimensional algebra of the form KeI, for I a nilpotent ideal - 90 given a parts can be the even a T-model structure for any K-DCT, cf. Kock [6].)&#x3E; Thus equation must hold since A is of the two sides of the T-model, while the odd parts are equal trivially. T, any Grassmann algebra any Z/2-graded sub-quotient of PROPOSITION 1.5. For any K-DCT over S(T)-model, a algebra over is as K is an Grassmann K. PROOF. All operations of S (T) are defined on any Grassmann algebra K over (or any subquotient) by the formulas used in defining Grassmann instantiations. Likewise any equation mu’st hold in any Grassmann algebra (and hence in any subquotient) because the equations of S(T) were taken to be precisely those which hold in all Grassmann instantiations! DEFINITION 1.6. A graded weil algebra over K is a finite dimensional unital graded-commutative algebra A, of the form (KOAB)OAF, where the part in parentheses is the bosonic grade, and As-OAF is a nilpotent ideal. A Weil algebra over K is a finite dimensional unital commut- algebra A of the form K4oIA where IA is a nilpotent ideal and K spanned by the multiplicative identity. We identify Weil algebras with those graded Weil algebras with trivial fermionic grade. ative is We then have PROPOSITION 1.7. Any graded structure for any K-DCT T . PROOF. All graded finitely generated As hint Weil Veil algebra has algebras are isomorphic algebras. to an S(T)-model subquotients of Grassmann to the reader where all these preliminaries are define a "topos of right superspaces" for any DCT T: namely, Sets fgs(T)-mod, where fgS(T)-mod is the category of finitely generated S(T)-models in Sets, and the algebra from which "supermanifolds" will be built inside the topos is the object named by the. forgetful functor U. While the algebra U has many of the good properties we are seeking, this topos lacks some of the geometric flavor we want, so some more preliminaries are in order. leading, a we could now 91 2. GEOMETRIC NOTIONS THEIR SUPERFICATIONS, In this section we ASSOCIATED shall at first TO consider DCT’S DCT’s superfications on an equal footing, and denote theories of by T unless otherwise clear from context. The next three are AND and their either type definitions extensions of notions found in Kock [6]. DEFINITION 2.1. A point of a T-model A is an algebra homomorphism p: A -&#x3E; K. We denote the set of points of A by pts(A). A model is point determined if DEFINITION 2.2. If A and B are T-models, (2-sided) ideal in A, A, by (resp. A{M-1}, AOT B, A{x}) we mean a T-model equipped with a T-model homomorphism A e A/I such that I is mapped to 0, and universal among such (resp. a T-model equipped with a T-model homomorphism A A A {M-1} such that all elements of E are mapped to invertible elements and universal among such; the coproduct in T-mod of A and B; the coproduct of A and the free T-model on 1 and E a subset of generator then I a A/I (x)). Note by standard exactness properties algebraic completeness we introduce the notion of germ-determined algebras, although we use it little in the sequel. that of all the above must theories. For DEFINITION 2.3. If p: A -&#x3E; K is a point exist of A, let Ap = A {Mp-1} is called the algebra of germs at p. Let be the canonical map. An ideal I is germ -determined if then (-) p : A -&#x3E; Ap germ-radical, I", of I is the smallest germ-determined ideal containing I. A T-model is germ- determined if its 0-ideal is germThe determined. 92 NOTE. The full T-models is tive A 4 A/0. subcategory of germ-determined subcategory, the reflection being given by It is claimed (Kock [6D that germ-determinedness is version of "geometrically interesting". reflec- a a rigorous construct a large number of interesting models for theories. The first observation to be made is that for superfied any DCT T there are many point-determined T-models: for the theory of C-algebras, the coordinate ring on any algebraic variety; Ralgebras, the coordinate ring of any algebraic variety whose reallocus is Zariski dense in its complex-locus; for C--algebras, the We can now our of Coo-functions on any smooth (paracompact, Hausdorff) manifold; and for the theory of homorphic functions, the ring of holomorphic functions on any Stein space, are all points determined models of their respective theories. Moreover in each case, the algebraic points ring correspond to the of the geometric points Armed with this, the following (germ-determined) S ( T ) -models: PROPOSITION 2.4. If A is a undrlying space. proposition provides point-determined T -algebra a wealth of (for T a K- pointdetermined (and hence a fortiori germ-determined) S(T)-model. If, moreover, W is a graded Veil algebra over K, then A9KW is a germ-determined S ( T ) -model (and is in fact the coproduct of A wi th the trivial grading and W). DCT), th en A with the trivial grading of Proposi ti on PROOF. That A with S ( T ) -model follows model the trivial immediately grading from homomorphisms are precisely the is point 1.4 is a determined as a that S ( T ) with the trivial observation between T-models equipped T-model homomorphisms. That AOKW has an S (T) -model structure follows from the fact that to verify the equations it suffices to verify them pointwise (i.e., after passing along pOkW), and in W they hold by Corollary 1.6. That AOKW is a coproduct in S (T) -mod follows from a standard result: if a (co) limit, in the category of models for a weaker theory, of models for a stronger theory is a model of the stronger theory, then it is a (co) limit in the category of models for the stronger theory. In light of this, we drop the subscript on the tensor product. To see that AOW is germ-determined, note first that A is a fortiori germ-determined. Now given any point p: AOW 4 K, this grading 93 factors through unique nilpotent the map AOV e A with ideal of W, giving a where Aol. kernel point p: A e K. Now Iw the is analysing this factorization shows that E, is in fact But since Iw is equivalent nilpotent, AoIw is also, inverting Zol. Thus if to for all points for all points p. and hence, p, this is Thus this is since A is and thus equivalent and thus is to equivalent germ-determined, to to a - 0 for all w ex.., = E Iw, , Thus AOV is COROLLARY germ-determined. 2.5. If M is smooth a (Coo)&#x3E; (resp. a Stein (resp. C), then (resp. SHol-model). manifold manifold), and W is any graded Weil algebra Coo (M)OW inverting E, (resp. Hol (M) OW) is an SCoo-model over R The reader will note that, in particular, W may be taken to be a finitely generated Grassmann algebra, in which case these algebras will be useful in our consideration of the relationship between Kostant’s graded manifolds and the topoi we will construct. The reader will also note that the restriction to Stein manifolds in the super-holomorphic case is related to the failure Theorem for graded holomorphic manifolds. (Although it of this restored in the scope paper, we holomorphic graded manifolds whose body Finally with respect define we to a the is a Batchelor’s is outside the Batchelor’s Theorem is restricts one’s attention to Stein manifold.) conjecture case of if category DCT, T. 94 that one of manifolds which are "good" DEFINITION 2.6. A (paracompact) manifold is a T -manifold if it is equipped with an atlas such that the transition functions between the charts are restrictions of T-operations to the coordinate chart. A continuous map between T-manifolds is a T - manifold map if its restriction to the intersection of any chart in the source, and the inverse image of a chart in the target, is the restriction of a T- operation. Given a T-manifold M, the coordinate T-algebra T(M) is algebra of global sections of the sheaf of T-algebras associated to the presheaf of T-algbras whose sections on coordinate charts are the T-operations restricted to the chart. (Note T &#x3E; is a contravariant functor from T-mf to T-alg). By abuse of notation, we also denote the sheaf of T-algebras described above by T(M), it being clear from context whether an algebra or sheaf is meant. is A T -manifold M (T-)complete if the "evaluation map" IMi e Pts (T(M)) is epi. M is (T-)separated if the evaluation map is the monic. A T-manifold M is good if it is complete and separated. M is if every cover by open sub-T-manifolds admits a refine- locally good ment by good good, good, 3. open sub-T-manifolds. The reader will note, for example, that all Coo-manifolds are Coowhile for complex analytic manifolds, only Stein spaces are but every analytic manifold is locally good. TORO I 3.1. 0F SUPERSPACES. Construction General Properties. topos E S(T) given as Shv(G,J), of S (T)-alg consisting of all S(T)-algebras of the form T (M)OW where M is a good T-manifold, and W is a graded Weil algebra; and J is the Grothendieck topology induced by T ( )&#x3E; of all open coverings of T-manifolds. In the case where T is Coo-alg, we call this the "super-Dubuc topos"; in the case where T is the theory of holomorphic functions, we call this the "super-Stein topos". and For any DCT, we consider the where G is the full subcategory DEFIBITIOH 3.1.1. For a commutative ring k in a fixed base topos S, a superlined toposlk (resp. lined toposlk) is an S-topos E equipped with a graded commutative ring object (resp. commutative ring object) R satisfying 95 Ll. For any graded Weil algebra/k canonical map ROW -&#x3E; Rspec (w) transpose to (where Spec (W)&#x3E; isomorphism. and L2. Spec(V) algebra) V (i.e., is is HomR-alg(ROW, R), Weil interpreted algebra/k) W, internally), a the is graded Weil algebra (resp. right adjoint, cf. Yetter (11J). an Weil for every tiny ( )" has (resp. THEOREX 3.1.2. For any k-DCT Es(T) js a superlined topos over k when equipped with R, the sheafification of the forgetful functor. PROOF. L1 follows by the proof of Kock [6], Theorem I I I .1.2, when that proof is taken at its full generality. To see this, it is necessary to verify that the tensor product of graded k-algebras is in fact the coproduct in the category of k-algebras, and (for DCT’s other than the theory of k-algebras) the observation concerning colimits for models of different theories made in the proof of 2.4. For L2, note that R is representable, and hence by a result of Bunge [3] the representable presheaf is tiny coproducts) in Setsc*P. The result then follows (since the site has the sufficient condition in Yetter [11]J for sheafification to preserve tininess. We now ,turn to is intrinsic in the without [11] : regard a way of sense recovering purely that it to how that topos can was from bosonic spaces which be done in any superlined topos constructed. Recall from Yetter DEFINITION 3.1.3. An object X is A-discrete whenever for all objects Y and all maps f: YxA -i X, f factors through the projection onto Y (i.e., "Maps from A to X are all constant", interpreted internally). DEFINITION 3.1.4. Spec(W)-discrete fermionic grade. An objects in Es(T) is pure for all graded Veil algebras W bosonic if generated by it is their PROPOSITION 3.1.5. The full subcategory of purely bosonic objects is a reflective, coreflective subtopos of ES(T), which we denote BOSs(T). We denote the reflection by body ( ), and the coreflection by cobody( ). 96 PROOF. Immediate by Intuitively, results in Yetter [11]. these functors correspond to the two ways to pass from a graded commmutative algebra to a commutative algebra: body( ) is quotienting by the ideal generated by the fermionic grade; cobody( )&#x3E; is cutting down the bosonic part. Care is required in interpreting this, since "bosonic part" means here not the bosonic grade, but the part of the algebra to which no odd element can be mapped under any morphism in the topos (internally!). Regretably, the cobody is the more interesting topos theoretically, use in applications. As little understood to be of interest, we and is an yet as example too of its prove: PROPOSITION 3.1.6. cobody(R) is a line in BOSs(T). Recall from Yetter [11] that the discrete reflection is an adjoint to the inclusion of discretes as functors enriched over the topos of discretes. Thus for any purely bosonic Weil algebra W we have PROOF. Note in the middle isomorphism in the isomorphism that follows cobody from is idempotent. The last ROW Cresp. sequence cobody CR)9W) is isomorphic to Rn (resp. cobody(R)n) for n = dim (W). while cobody C ) is limit preserving. (Warning: cobody( ) does not in general preserve colimits (e.g. 0) - it does so in this case only because these instances of 0 can be canonically re-expressed as limits, which are preserved.)&#x3E; the fact that Although the intrinsic nature of these constructions suggests that their study is fruitful, the cobody construction depends upon the little understood, but powerful, properties of tiny objects (see Yetter [11]), so that some fundamental work is required before this construction can be properly applied. We turn therefore to a construction of a subtopos of "bosonic" objects, which is extrinsic in the sense that it is carried out at the level of defining sites: DEFINITION topos Es(T) 3.1.7. The subtopos of bosonic sheaves- BShs T&#x3E;, in the is the topos Shv (G, K), where G is as in the definition 97 of Es T&#x3E;, and K is of covers considered as the form As topology generated by is the J and all one bosonic grade object of A, trivially graded S (T)-algebra). a topoi our As (where -&#x3E; A following proposition The between the establishes then the relationship of superspaces and standard models for SDG: PROPOSTIION 3.1.8. If Es(T) is the super--Dubuc topos Cresp. superStein topas), then BShs (T) is equivalent to the Dubuc topos (resp. the Stein topos), and if R. is the sheafification of R, and is the usual line in the Dubuc topos (resp. the Stein topos). PROOF. The sites of definition are equivalent. (The defining site for the latter topos is included in the defining site for the former, and every object in the larger site is canonically covered by an object in the smaller.) For the conclusion about the superline, observe that R and Re are representable, and that R’s representing object in the site is covered by the representing object for Rat Moreover, it is easy to see that RB is carried to the usual line, in the Dubuc (resp. Stein) topos by 3.2. Graded Adapting manifolds, we the a is an (Z/2-)graded open is of a pair (X,A), graded algebras over where X X such of X by T-manifolds, {Ui} i e I such that finitely generated Grassmann algebra. Maps cover A some T-manifold is sheaf a defined in the obvious way. We let GT-Mf X A and A (Ui) = T (A)OA, for are and topoi of superspaces. definition [7] of graded manifolds to T- make: T-manifold, that there of sites. manifolds Kostant’s DEFINITION 3.2.1. A is equivalence locally good, manifolds folds"). We with X can now relation between category of graded T-manifolds with let GT-Mfo denote the category of graded and A =T(X)OA ("good trivially graded mani- denote the and good state and graded prove a manifolds and comparison theorem showing our topoi of superspaces: THEOREX 3.2.2. There is a functor i : GT-Mf 4 Es T&#x3E; composite functor r7L: GT-Mfo -&#x3E; EsT&#x3E; (r being the 98 the extending the global section functor, y the Yoneda embedding Into the presheaf topos, and L the sheafification functor) and satisfying : 0. i is full and faithful,. 1 1. pull backs wh i ch are transversal pull backs body. covers to epilnorphjc families. is a superline. preserves all when restricted to the 2, 1 carries open and 3. i(kOA (i))= R PROOF. We begin by noting that if 1 extends FyL, then we have already shown 3, since (k,T(k)OA(i)) is in GT-Mfo. We next note that ryL satisfies 2, by construction of the topology in the site of definition, while ryL satisfies 1 by applying results of Kock 161 once it is noted that (M,T (M)OA) is isomorphic to the product (M,T (M))x(*, A) and that r, y, and L all preserve products. To see that FyL satisfies 0, it suffices to examine r, since y is full and faithful and the topology in question is subcanonical. For f, fullness together manifolds functions, and faithfulness follow from the product with the observations that on (*, A) a map is entirely determined while "goodness" allows by its us to decomposition, of graded T- behaviour on the ring of the classical proof imitate that C-( )&#x3E; is full and faithful for any T. To extend ryL to all of GT -Mf, note that any locally good is the colimit of its good trivialT-manifold graded canonically izations, that is of a canonically chosen diagram in GT-Mfo. We let i be the result of applying ryL to this diagram, then taking the colimit in EsT&#x3E;. Note that this extends fyL, since it agrees with FyL on GT-Mfo, since here the diagram of good trivializations has a terminal object. Now since 1 and 2 are local in nature, the colimiting construcwill preserve them. For 0 note that the image of GT- Mfo generates EsT&#x3E;, and thus I must be faithful, while fullness follows from 2 by passing to a good trivialization of the target, and then to a good trivialization of the source which refines its preimage. tion Thus the theory as "super-Dubuc topos" plays the topos does for classical the Dubuc 3.3. same "super" geometry. role in the differential Formal spermanifolds. Although all objects in the topoi EsT&#x3E; can be regarded as "superspaces", they do not all possess manifold-like properties. Two approaches may be taken to isolating "formal supermanifolds". The first is essentially classical: choose model objects and define manifolds as those objects which "look locally like the models". The 99 second purely synthetic: determine what properties of manifolds to the problem at hand and consider those objects which satisfy them (having shown that those objects which intuitively "should" be manifolds satisfy the properties). We begin are is essential with the former: The obvious notion of supermanifald arises of the form RB FxRFo, where by taking as model objects all objects then cover considering all objects X such that there is a formal etale {Ui} i E I by formal etale subobjects of the model ob jects, where: DEFINITION 3.3.1. A map f: X -) Y is formal etale if subobject A of it -(0) C Rn, containing 0, for any n, the for any tiny diagram is a pullback. "supermanifold" is sufficient to include the graded manifolds, but fails to capture the have good local behaviour and are one of which "superfunction spaces" the points of the synthetic approach. This internal notion versions of of For the purely synthetic approach, we wish particularly interesting Weil algebra spectra: to distinguish some DEFICIT ION 3 .3 .2 . Let D (p, q) = {(x1, ... , Xp1 01, ...0q) Xi I bosonic, Ok fermionic, Dk(p,q)={(x1,...,xp, 01,...04) any (k+l)-fold product 100 I x1 x3= x, xi0k= OkO, = 0) C Rp+q, bosonic, 8k fermionic, of the xi’s and 0k’s is 0) C Rp+q, Note that D (0,1)&#x3E; = RF since all fermionic elements are 2-nil- potents. infinitesimal of versions can now formulate "super" the of and Both (see [6]). Kock W"" following linearity "Property definitions are to be read internally, so that maps are to be taken We as generalized elements of the appropriate function objects. DEFINITION 3.3.3. An object in a superlined linear if given any family of maps topos is infinitesina]17 such that uniquely there exist (resp. i(o,J) is inclusion by setting (resp. fh fermionic) to 0. where i(i, o) except the jth bosonic DEFINITION 3.3.4. An object satisfies D (1,0) FxD (0,1)a -1 X such that Property W(p,q) all coordinates if for all maps 7: there exists uniquely t: D (E,A) A X such that where (Ë,X) = (1,0) if q is PROPOSITION 3.3.5. R is W (p,q). PROOF. As from Ll. in the even and (E, A)= (0,1) infinitesimal7 ungraded case (in 101 if q is odd. linear and satisfies Kock 161)&#x3E; this follows Property readily THEDREX 3.3.6. class T’he of infinitesimally objects linear (resp. objects satisfying Property W (p,q)) is closed under: Cl. formal etale subobjects, C2, limit, C3. exponentiation by arbi trary objects, passing t o factors of products, arbitrary coproducts. C4 . and C5. PROOF. C3 is , immediate from the internal nature is immediate from the universal and the that conditions property of limits in the conclusions of the definitions. For uniqueness neighborhood factors involved map every the of involved, while C2 of the image of 0 in the through Cl, note formal any etale objects involved. to AxB in the hypotheses of the defuniquely expressed as a pair of maps, one to A, one to B, which each satisfy the hypotheses; the unique pair, each of which is given by the existential part of the definitions, defines a map to AxB which has the same property. For initions C5 object C4, note that the maps are follows from involved. the tininess (Note: tininess existential conditions will not in COROLLARY connectedness)&#x3E; hence (and gives in the definitions of the preservation of by arbitrary colimits, the the but general give uniqueness.) 3.3.7. infini tesimally Formal linear and supermanifolds cln the sense above) satisfy Property W (p,q) for all (p,q). are It is in fact these two properties: infinitesimal linearity and (for certain Property W(p,q) p and q) which give most of the "classical" properties of the tangent bundle once the correct definition of that notion is introduced. Two reasonable notions present themselves. DEFINITION 3.3.8. The total tangent bundle of X is the object over X given by The bosonic The module latter for concentrate tangent of DeWitt our tangent bundle, these bundle is the object corresponds supermanifolds, and attention on the more the total tangent bundle: 102 over X given by the tangent properties. We genuinely "super" notion of more has or less similar to THEOREM 3.3.9. If X is infinitesimally linear, then the total tangent bundle is a bundle of R,R- bimodules over X, satisfying moreover where a E each case. PROOF. By R, x c XD(1,1)p, and infinitesimal linearity valued denotes the 0-1 I I we have an grading in isomorphism this with the map XA: XD12.21 4 XD(1,1) gives the addition on tangent bundle. Verification that this gives a fibrewise abelian group structure is essentially as in Kock 161. The bimodule structure is given by Composing total the Both distributivity graded commutativity (D (1,1) being as and COROLLARY 3.3.10. For M fields on M, is a graded associativity follows from subobject of R). easy to verify, graded commutativity are the infinitesimally linear, Property W(p,q) can now existing classically be used on VectCM): to structure. THEOREM 3.3.11. If M is in.f.initesjmally perties V(2,0), W(1,1), W(0,2), then graded Lie algebra given gradewise by a of R object of vector commutative RM-module. structure is the while over R, 103 wben provide the additional (graded) Lie algebra a linear and equipped with satisfies Pro- the operation which is the unique map given by Property W(p,q) c’a,ba and (a, B) are each one of (1,0) (1,0) if b+B is even and (0,1) if b+B is odd. wbere PROOF. An imitation of the Kock (61) suffices when the account. or such that (0,1), and (E,A) is argument due to Reyes graded commutativity of and Wraith (see R is taken into COROLLARY 3.3.12. If G is a group object, which is Infinitesimally linear and satisfies Properties W (2,0), W (1,1), V(0,2), then is a graded PROOF. Lie algebralR. Identify T.(G)&#x3E; and restrict the Lie with the algebra object of left invariant vector fields structure of Theorem 3.3.11. besides internal versions of finite dimensional exotic but physically interesting objects as the supergroups, internal versions of "super-loop groups" satisfy the hypotheses of the Corollary, and thus are included in the same synthetic constructions as the finite dimensional cases. Note that such Order 3,4, topos. and Integration i n the super-Dubuc Finally, we turn to superspace integration in the context of our models. Recall that superspace integration in other models of superspace (cf. Rogers [10] or Berezin t2» is carried out by treating bosonic and fermionic coordinates differently: bosonic variables are integrated classically, while fermionic coordinates are integrated according to the Berezin prescription: As noted in the superspace coordinates, introduction, it is the view of Batchelor that in bosonic is really a hybrid: integration differentiation in fermionic coordinates. integration 104 to the theory S(C.-alg), and the denote E. This will be necessary super-Dubuc topos, only to consider integration in bosonic parameters. For fermionic ones in any superlined topos we have: We restrict our attention which now now we DEFINITION 3.4.1. The Bereain integral a’’ pz, where 0:: RxR e RRF is given by and is invertible by map (RFf -&#x3E; L1 in the definition of R is the composite superline. that this definition is internal, and hence "smooth in parameters". It is also precisely the fermionic parameter version of synthetic differentiation: in the view of Batchelor, "Berezin Note integration is odd parameter differentiation". Proceeding on to bosonic parameter integration, Theorem 3.1.8 allows integration structure be precise: us on note that order structure and "classical" the Dubuc topos to the super-Dubuc topos. To to lift the THEOREX 3.4.2. R (resp. RB&#x3E; has two preorderings, and satisfying: 01. and are transitive. 02. is reflexive ; is irrefl ex,i ve. . x y 4 a y+z ; xy =&#x3E; x+ z z. 04. [x y A 0&#x3E;t]=&#x3E; xt yt; [x y A 0 t]=&#x3E; xt yt. 05. 0 1. 06. x 0 =&#x3E; x 0. 07. d nilpotent (0 d A d ; 0). 08 , x 0 =&#x3E; x invertible. 09. -1 (x 0) =0 (x. 010. x Inver-tible 4 Ex 011. CO x A x y]=&#x3E; 0 0 V 0 x]. y. PROOF. For RB this is a result of Kock (6) for the Dubuc topos. To extend the orderings of R, note that any element of R is of the form B (x)+F (x) for B (x) E RB and F (x) c RF. Let x ( y iff B (x) ( It is then easy to extension (the crucial always nilpotent). B(y) and verify that thing is to 105 x y 01-011 iff are B (x) BCy) . preserved by note that fermionic elements this are We denote by [0,1] the subobject of Re, {x I 0 E x ( 1?, and by (0,1) the subobject of R given by the same formula. Note that Except for the difficulty that we need our result to hold "smoothly in fermionic parameters" (i.e., for generalized elements given by fermionic objects), we could now just lift the integration from the Dubuc topos to give our superspace integration in bosonic parameters. Instead we must imitate the proof of the "Integration Axiom" for the Dubuc topos, and check that the resulting bosonic parameter integration commutes with Berezin integration in fermionic parameters. Before proceeding further, we must note: PROPOSITION 3.4.3. The functor i: GC*-Mf-&#x3E; E extends to a functor .from the category of graded CG’&#x3E; manifolds wi th boundary, so as to agree with the extension of the functor from smooth manifolds to the Dubuc topos to smooth manifolds wi th boundary. We continue to denote this extension by 1. COROLLARY 3.4.4. THEOREM 3.4.5. For a n y f E RCO,13 in E , there is such that g(D) = 0 and g’= f, where ( )’ denotes tiation in one bosonic parameter Ci.e., a-1 p2, unique g E RCO, 13 synthetic differen- where a a: RxR -&#x3E; RD(1, 0) is given by and a is invertible We denote g(x) by by L1 in the definition of (f (x) superline). dx. PROOF. Consider generalized elements f c RIO." of type 2(M)xSpec(V)&#x3E; for M a C’-manifold, and W a graded Weil algbra. We then have a sequence of natural correspondences: 106 dim(W)-tuples of maps i(Mx[0,1])&#x3E; dim(W)-tuples of maps Mx[0,1] -&#x3E; R 4 Re in E in the (equiv. category in Dubuc topos) of smooth manifolds with boundary. The passage to the Dubuc topos requires us to note that there are not global non-zero maps from any bosonic object to RF (equiv. the bosonic sheafification of RF is 1). We now integrate classically in each coordinate and reverse the sequence of natural equivalences to obtain the (generalized) element g, noting that each equivalence "preserves (bosonic) differentiation" in the evident sense. It is then PROPOSITION parameters integration. an easy consequence of cartesian closedness that: The value 3.4.6. (of possibly mixed A final note on of types) integration: the integrals independent of iterated is synthetic approach in makes clear "differentiatiuon backwards" aspect of integration lost in notions of integration applicable to superfields. why the Consider the definition of differentiation in f’ : R-&#x3E;R is the When a lined several the order of must be topos: unique function such that V x E R V d E D f (x+d)- f(x)= df’(x). superlined topos, and replace D by D (1,1)&#x3E; in the super setting), no such function exists in general: instead there is a unique function f’: R e M (1,1), where M (1 ,1) is the object of (1,1)-square supermatrices. It is thus impossible to identify functions with vector-fields on the superline by any "superEuclidean metric" and thus to identify integration of superfields with genuine anti-differentiation. (the we pass to a object of 2-nilpotents ACKNOWLEDGEMENTS. The author extends his thanks to the National Scince Foundation for support while the author was in residence at the Institute for Advanced Study, where this work was begun (grant #DMS-8610730 (1)), and to the Groupe Interuniversitaire en Etudes Cat6goriques for support while this work was completed. 107 BIBLIOGRAPHY. BATCHELOR,M., Graded manifolds and supermanifolds, in [9], BEREZIN, F.A., Differential forms on supermanifolds, Sov, J, Nucl, Phys, 30 (1979), 605-608, 3, BUNGE, M., Categories of set-valued functors, Ph.D. Thesis, Univ, of Pennsylvania, 1966 (unpublished), 4, DUBUC, E., Sur les modèles de la Géométrie Différentielle Synthé-tique, 1, 2, Top, et Géom, Diff, XX-3 (1979), 231-279, HOSKIN, D,A,J,, The Stein topos, in Categorical Methods in Geometry (ed, A. Kock), Aarhus Univ. 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