A Categorical Formulation of Superalgebra and Supergeometry
aa r X i v : . [ m a t h . AG ] F e b A Categorical Formulation of Superalgebra andSupergeometry
Christoph SachseMax Planck Institute for Mathematics in the Sciences,Inselstr. 22, D-04103 Leipzig, Germany
Abstract
We reformulate superalgebra and supergeometry in completely cat-egorical terms by a consequent use of the functor of points. The in-creased abstraction of this approach is rewarded by a number of greatadvantages. First, we show that one can extend supergeometry com-pletely naturally to infinite-dimensional contexts. Secondly, some sub-tle and sometimes obscure-seeming points of supergeometry becomeclear in light of these results, e.g., the relation between the Berezin-Leites-Kostant and the de Witt-Rogers approaches, and the precisegeometric meaning of odd parameters in supergeometry. In addition,this method allows us to construct in an easy manner superspaces ofmorphisms between superobjects, i.e., the inner Hom objects associ-ated with the sets of such morphisms. The results of our work relyheavily and were inspired by the ideas of V. Molotkov, who in [13]outlined the approach extensively expounded here.
Superalgebra and supergeometry are fields of mathematics which owe theirinvention to a physical concept: supersymmetry. In quantum field theory,fields of integer spin (called bosons) are described by commuting variableswhile those of half-integer spin (fermions) have to be anticommuting in orderfor the theory to produce meaningful results. It was Berezin who had theidea to combine these physically very differently behaved types of fields intoa single Z -graded space. Operators, as well, could then form Z -graded al-gebras, and their operation would have to preserve parity. The radically newoption that this formalism introduces is that one can also write down trans-formations which are proportional to odd parameters. This made it possibleto transform bosons into fermions and vice versa by means of fermionic pa-rameters. Supersymmetry is then nothing else than the statement that aphysical theory remains invariant under this type of transformations.In mathematical terms, this means that supersymmetry does not simplyconsist of the idea to use Z -graded spaces and algebras, but also to carry1ut all operations over a Z -graded base. Physical applications, in particularin gravity theory, very soon made it desirable to extend this concept frompure algebra to geometry. This geometry would have to include odd as wellas even coordinates and treat them on a completely equal footing. It wasfound out already in the seventies that this is indeed possible [8]. Again, thefull meaning of supergeometry only becomes visible by working over a basewhich is itself “super” in order to allow for odd parameters in all coordinatechanges. The resulting theory extends commutative to “supercommutative”geometry. Its noncommutativity is so tame (an actual noncommutativegeometer would probably not even call it noncommutative) that it still allowsto carry over almost all objects of classical geometry, but it also alreadyexhibits completely new phenomena.The correct description of supermanifolds required several technical tools(e.g., sheaf theory, point functors, working in families, etc.) which were newto physics at that time, and this fact has made the relationship betweenphysicists and mathematicians working on super questions at times a dif-ficult one. From a mathematical point of view, the ringed space approachcommon in algebraic geometry is the obvious and essentially the only wayto extend the ideas of superalgebra to the realm of geometry. Rings are thestarting point for the construction of any geometric space in algebraic ge-ometry anyway, and the transition from commutative to supercommutativeones can be handled in a completely natural and very elegant way in thisformalism. The price, however, is a considerable abstraction and technicalcomplexity.Therefore, various other approaches have been advocated, most promi-nently the Rogers-de Witt approach. These approaches mostly tried toavoid the technicalities of sheaves and point functors and tried to repro-duce a closer analogue of ordinary differential geometry. Many practicionersworking on supersymmetric field theories and string theory have found theseapproaches easier to handle, and they were sometimes claimed to be “moreadequate to physics” than the algebraic approach. The categorical formula-tion outlined below will allow us make the relation between these approachescompletely transparent. In particular, we will argue that the more concretelooking de Witt approach is indeed correct as long as one constructs every-thing functorially in the Grassmann algebra that one chooses (and infinite-dimensional Grassmann algebras are unnecessary). Although this fact seemsto have been rarely spelled out in the literature, this is the way the Rogers-de Witt geometry has been used in most applications. In this case, it isequivalent to the use of just one set of points of the corresponding pointfunctor of the super object under study. Any kind of topologization of theodd dimensions of a supermanifold, however, will produce faulty results,since it destroys functoriality with respect to base change.The categorical approach should not be understood as simply one moreway to phrase the same old results, but rather as a complement to the2inged space approach. It is based on the exploitation of a tool which isstandard in algebraic geometry, namely the functor of points supplied bythe Yoneda Lemma, which describes an object of a given category by thesets of morphisms of other objects of that category into it. The usefulnessof this method was, of course, well known to the founders of the ringedspace approach [8], [2], [12], and was used, e.g., for the definition of super-groups [9]. From its definition alone it is, however, often completely unclearhow one could get a manageable description of this functor which allowsconcrete calculations and constructions. V. Molotkov in [13] outlined a pro-gram that reformulated superalgebra as well as supergeometry entirely inthis categorical framework and also proposed a way to make the functor ofpoints really usable, namely by presenting a countable set of generators forthe category of supermanifolds. Unfortunately, his preprint [13] containedno proofs, and is not easily accessible for more practically or physically ori-ented researchers. In this work we carry out most of the program proposedby Molotkov and develop supermathematics from a categorical perspective.Similar ideas have been investigated in [16], [18].Our original motivation to investigate these questions was the need fora practicable model for infinite-dimensional supergeometry. Ringed spacesbecome inadequate in infinte dimensions, and one has to resort to conceptslike functored spaces. This is not a super problem, but occurs as well in ordi-nary infinite-dimensional geometry [5]. The categorical construction solvesthis problem for supermanifolds. As an example it will be used in a subse-quent paper [15] to explicitly construct the diffeomorphism supergroup of asupermanifold M . Acknowledgments
I thank the Klaus Tschira Stiftung and the MPI for Mathematics in theSciences for financial support. The results of this work were in large partsinspired and some already announced in Vladimir Molotkov’s preprint [13].Besides these ideas I owe him thanks for continued advice and correctionswhile trying to forge this bunch of ideas into a neat piece of mathematics. Iam also grateful to J¨urgen Jost, Dimitri Leites, Guy Buss and Brian Clarkefor encouragement, advice and valuable criticism.
This section contains a very brief review of the key ideas and constructions ofsuperalgebra and supergeometry. For in-depth (mathematical) treatmentsof the matters in this section see, e.g., [3], [9], [17].3 .1 Linear superspaces and superalgebras
The elements of Z = Z / Z will be denoted as { ¯0 , ¯1 } . The field K denoteseither R or C . We begin by recalling that a superring R is simply a Z -graded ring, i.e. R = R ¯0 ⊕ R ¯1 , and that a morphism of superrings is amorphism of graded rings. Definition 2.1.
A module M over a superring R is called a supermodule,if it is Z -graded, M = M ¯0 ⊕ M ¯1 , (1) and if R ¯ i · M ¯ j ⊆ M ¯ i +¯ j . The submodule M ¯0 is called even, M ¯1 is called odd.A morphism φ : M → M ′ of R -supermodules is a morphism of R -moduleswhich preserves the grading. We denote by
SMod R (resp. R SMod ) the category of right (resp. left) R -supermodules. Note that any ring R can be considered as a superring byjust setting R ¯0 := R and R ¯1 := 0.An element m ∈ M ¯ i is said to be homogeneous of parity p ( m ) = i . Aninhomogeneous element is said to be of indefinite parity.A supermodule over a field K is called a K -super vector space. Thecategory of K -super vector spaces will be denoted by SVec K . Particularlyimportant are the standard super vector spaces K m | n , which have m evenand n odd dimensions. Definition 2.2. A K -superalgebra A is a K -super vector space endowedwith a K -bilinear morphism µ : A × A → A. (2) The algebra A is called supercommutative if for all homogeneous elements a, b , one has µ ( a, b ) = ( − p ( a ) p ( b ) µ ( b, a ) . (3) A is associative, resp. with unit, if it is associative, resp. has a unit, as anordinary K -algebra. Lie superalgebras are defined completely analogously. Expressions like(3) are extended to inhomogeneous elements by linearity. It is clear that if aunit exists, it has to be even. If no confusion can arise, multiplication in analgebra will be denoted by either a dot (like a · b ) or by juxtaposition (like ab ). In the following, we will assume that all superalgebras are associativeand have a unit.For a supercommutative algebra, every left module can be made a rightmodule by defining m · a := ( − p ( a ) p ( m ) a · m (4)4or all homogeneous elements a ∈ A and m ∈ M , and extending this defi-nition by linearity. From now on, we will restrict ourselves to supercommu-tative algebras with unit. Consequently, we will just speak of “modules”,all of which are defined to be left-modules whose right module structure isgiven by (4).Direct sums, tensor products and Hom-spaces can be defined in an ob-vious way for supermodules. Let A be a supercommutative K -superalgebraand M, N be A -modules. Then( M ⊕ N ) ¯ i := M ¯ i ⊕ N ¯ i , (5)( M ⊗ A N ) := ( M ⊗ K N ) /I, (6)where I is the ideal generated by elements of the form ma ⊗ n − m ⊗ an ,and ( M ⊗ K N ) ¯ i := M ¯ j +¯ k =¯ i M ¯ j ⊗ K N ¯ k . (7)Finally one habitually setsHom A ( M, N ) ¯ i := { f : M → N | f ( M ¯ j ) ⊂ N ¯ j +¯ i } (8)where on the right hand side, f is meant to be an ordinary module homomor-phism between M and N . It is important to note here that Hom A ( M, N )does in general not coincide with Hom A ( M, N ): the former is the set of mor-phisms of supermodules, which by definition preserve parity. This set doesnot carry any natural supermodule structure itself. The latter is, strictlyspeaking, the inner Hom object associated to this set. One always hasHom A ( M, N ) ¯0 = Hom A ( M, N ), that is, the inner Hom object extends theset of morphisms of supermodules by adding parity reversing morphisms.The distinction between the two objects may seem overly picky at this point,but it will turn out that inner Hom objects play a particularly importantrole in supermathematics. In supergeometry, e.g., as diffeomorphism super-groups, their construction is by far less obvious than in superalgebra.The above definition of the tensor product together with the followingchoice of the commutativity isomorphisms c V,W : V ⊗ A W → W ⊗ A V (9) v ⊗ w ( − p ( v ) p ( w ) w ⊗ v. make SMod A a braided tensor category. This choice of the braiding encodesthe so-called sign rule which asserts that whenever two neighbouring factorsin a multiplicative expression involving elements of supermodules are inter-changed, one picks up a factor ( −
1) and thus is the key difference betweensuperalgebra and ordinary algebra.The braiding isomorphisms supply the notions of symmetric and exteriorpowers as well as those of symmetric and antisymmetric maps in the usual5ashion: one sets Sym n ( V ) to be the quotient of V ⊗ n divided by the action ofthe permutation group S n described by c V,V . Analogously, ∧ n ( V ) is definedas V ⊗ n divided by the action of S n times its sign character. Clearly, for V = V ¯0 ⊕ V ¯1 , Sym n ( V ) = n M k =0 Sym k ( V ¯0 ) ⊗ ∧ n − k ( V ¯1 ) , (10)where Sym and ∧ on the right hand side denote the ordinary operations onvector spaces. Symmetric (resp. “supersymmetric”) multilinear maps arethen morphisms f : V ⊗ n → W which are invariant under the action of S n by c V,V . In other words, they descend to well defined maps f : Sym n ( V ) → W .We will denote the set of supersymmetric morphisms between V ⊗ n and W by Sym n ( V ; W ). The sets Sym n ( V ; W ) are again the even subspaces ofinner Hom objects Sym n ( V ; W ). Definition 2.3.
Let R be a superring. The change of parity functor is thefunctor Π :
SMod R → SMod R (equivalently for left modules) which assignsto a supermodule M = M ¯0 ⊕ M ¯1 the supermodule Π( M ) with (Π( M )) ¯0 = M ¯1 and (Π( M )) ¯1 = M ¯0 . Parity reversal has to be a functor since any morphism between twosupermodules has to preserve parity. One clearly has Π( K ) = K | and forany K -super vector space V , K | ⊗ K V ∼ = Π( V ) . (11) A particularly important role will be played by the finitely generated Grass-mann algebras Λ n , i.e., the free commutative superalgebras on n odd gener-ators. In particular, we will use the category Gr , which contains exactly oneobject Λ n for each integer n ≥ n .Note that the field K = Λ is the null object of Gr : the terminal mor-phisms ǫ Λ n act by removing all odd generators, while the initial morphisms c Λ n inject K into Λ n .We will write Gr and Λ n both for the real and complex Grassmannalgebras and only distinguish them when the ground field really matters. Proposition 2.4.
Let Λ n , Λ m be the Grassmann algebras over K on n and m generators, respectively. Then there exists an isomorphism of K -vectorspaces Hom Gr (Λ n , Λ m ) ∼ = K n ⊗ K Λ m, ¯1 . (12)6 roof. Let ξ , . . . , ξ n be the free generators of Λ n . A morphism φ : Λ n → Λ m is a homomorphism of K -algebras which preserves parity. Thus φ (1) = 1,and φ is uniquely determined by choosing the images φ ( ξ n ) of its generatorswhich have to lie in Λ m, ¯1 . Setting V = Span K ( ξ , . . . , ξ n ), we can writeHom Gr (Λ n , Λ m ) ∼ = Λ m, ¯1 ⊗ K V ∗ , where V ∗ is the dual space of V . Here, again, we will only very briefly review the key ideas of supergeometry.See [9], [17], [3] for more complete treatments.Supergeometry is based on the idea to replace the commutative rings of“functions” which make up the structure sheaves of commutative geometricspaces by supercommutative ones. In contrast to general noncommutativegeometry, this works out within the classical framework of algebraic geome-try, because almost all crucial tools, like, e.g., localization, are amenable tosupercommutative rings.
Definition 2.5.
A locally superringed space M = ( M, O M ) is a topologicalspace M endowed with a sheaf O M of local supercommutative rings. Amorphism of superspaces is a morphism of locally ringed spaces which isstalkwise a homomorphism of supercommutative rings. The structure sheaf O M contains a canonical subsheaf of nilpotent idealsgenerated by odd elements: J = O M , ¯1 ⊕ O M , ¯1 . (13)The superspace M red = ( M, O M / J ) is then purely even and possesses acanonical embedding cem : M red → M . (14)The space M red is called the underlying space of M . In general, M red canitself be a non-reduced space. The “completely reduced” space is denotedas M rd . In this work we will not consider non-reduced underlying spaces,so we always assume M red = M rd .A particular role is played by the model spaces R m | n and C m | n , whichare defined by R m | n := ( R m , C ∞ R m [ θ , . . . , θ n ]) (15) C m | n := ( C m , O C m [ θ , . . . , θ n ]) (16)where C ∞ and O denote the sheaves of smooth and holomorphic functions,respectively, and θ , . . . , θ n are odd generators, which freely generate thestructure sheaf of these superspaces over C ∞ and O , respectively.7 efinition 2.6. A real, resp. complex, supermanifold of dimension m | n isa superringed space which is locally isomorphic to R m | n , resp. C m | n . Thus, strictly speaking, a super vector space is not a supermanifold.Each super vector space V gives rise to a supermanifold V = ( V ¯0 , C ∞ V ¯0 ⊗∧ ( V ¯1 )) (here ∧ ( V ¯1 ) denotes the ordinary exterior algebra over V ¯1 ), but thesetwo objects behave rather differently. For example, the underlying set-theoretical model of V is a ( m + n )-dimensional space, while that of V is only a m -dimensional one.Together with their morphisms as superringed spaces, smooth finite-di-mensional supermanifolds form a category which we will denote as FinSMan .Sections of the structure sheaf O M of a supermanifold M are calledsuperfunctions. Each superfunction can locally be expanded into powers ofthe odd generators: F ( x , . . . , x m , θ , . . . , θ n ) = X I ⊆{ ,...,n } f I ( x , . . . , x m ) θ I , (17)where the sum runs over all increasingly ordered subsets, θ I is the productof the appropriate local odd coordinates, and the f I are ordinary smooth(resp. holomorphic) functions. The value of F at x is defined to be f ( x ),and obviously does not determine F , in contrast to ordinary geometry. A finite-dimensional supermanifold whose underlying man-ifold is a one-point topological space is called a superpoint.
Superpoints will play an important role later on, analogous to that of thespaces Spec K for ordinary geometry. From the above discussion it is evidentthat superpoints are the supermanifolds associated to purely odd super vec-tor spaces, i.e., to those of dimension 0 | n . Together with their morphisms assupermanifolds, finite-dimensional superpoints form a category SPoint whichis a full subcategory of
SMan . Proposition 2.8 (see also [13]) . There exists an equivalence of categories P : Gr ◦ → SPoint . (18) Proof.
Define the functor P on the objects Λ n of Gr ◦ as P (Λ n ) := P n := Spec Λ n = ( {∗} , Λ n ) . (19)To every morphism ϕ : Λ n → Λ m of Grassmann algebras, assign the mor-phism Φ = (id {∗} , ϕ ) : P m → P n . (20)8o see that this functor establishes an equivalence, note first that it is fullyfaithful: on the set of morphisms it is a bijection from Hom Gr (Λ n , Λ m ) toHom SMan ( P m , P n ). The last property to check is essential surjectivity, i.e.,every superpoint has to be isomorphic to one of the P n . This is clear from thefact that since a superpoint is a supermanifold its structure sheaf must be afree superalgebra on n odd generators. Since any such algebra is isomorphicto Λ n , the assertion is proved.This allows us to restrict our attention to the supermanifolds Spec Λ n when talking about superpoints. As a first step we will translate linear and commutative superalgebra intothe language of the functor of points. This construction can be viewed as asystematic treatment of the so-called even rules principle, which is a way todo superalgebra without having to handle odd quantities [3], [17].We will use the following notational conventions throughout the rest ofthis paper: if C is an object of a category C , we write simply C ∈ C . Thecategory of functors C → D between the small category C and an arbitrarycategory D will be denoted as D C . For in-depth accounts of the categoricaltools used below, see [7], [10], [11]. R and C In the following, we will almost exclusively be concerned with the functorcategories
Sets Gr ⊃ Top Gr ⊃ Man Gr , that is, covariant functors from the category Gr of Grassmann algebras de-fined in 2.1.1 into the category of sets, topological spaces and smooth Banachmanifolds, respectively.For K = R , C , define a functor K ∈ Sets Gr by K (Λ) := (Λ ⊗ K K ) ¯0 = Λ ¯0 , (21) K ( ϕ ) := ϕ (cid:12)(cid:12) Λ ¯0 (22)for ϕ : Λ → Λ ′ a morphism in Gr , the category of Grassmann algebras overthe field K . Clearly, K is a commutative ring with unit in Sets Gr : each K (Λ)has a commutative ring structure inherited from Λ ¯0 , with a unit induced bythe unique morphism Λ → K .The rings K will replace the ground field K in the categorical definitionof super spaces. 9 .2 K -modules in Sets Gr In this section we will introduce a particular class of K -modules in Sets Gr ,which are the avatars super vector spaces over K in the categorical formu-lation.Let V be some real or complex super vector space. We define a functor V ∈ Sets Gr by setting V (Λ) := (Λ ⊗ K V ) ¯0 = (Λ ¯0 ⊗ V ¯0 ) ⊕ (Λ ¯1 ⊗ V ¯1 ) (23) V ( ϕ ) := ( ϕ ⊗ id V ) (cid:12)(cid:12) V (Λ) for ϕ : Λ → Λ ′ . All sets V (Λ) are naturally Λ ¯0 -modules, and thus, V is a K -module.Let f : V × . . . × V n → V be a multilinear map of K -super vector spaces.To f , we assign the functor morphism f : V × . . . × V n → V n whosecomponents f Λ : V (Λ) × . . . × V n (Λ) → V (Λ) (24)are defined by f Λ ( λ ⊗ v , . . . , λ n ⊗ v n ) := λ n · · · λ ⊗ f ( v , . . . , v n ) (25)for all λ i ⊗ v i ∈ V i (Λ). All maps f Λ are clearly Λ ¯0 -linear, hence f is a K - n -linear morphism in Mod K ( Sets Gr ).Given some commutative ring R , we will denote the set of R - n -linearmorphisms M × . . . × M n → M by L nR ( M , . . . , M n ; M ). L nR is naturallyan R -module itself, and if the M i , M are supermodules, L nR it is identical tothe inner Hom-object, i.e., a supermodule itself.So, in particular, L n K is a K -module which implies that it is a K -module,because Λ ¯0 -linearity always entails K -linearity. It turns out that the K -module structures on L n K , ¯0 and L n K coincide as a consequence of functoriality. Proposition 3.1 (see also [13] ) . The assignment f f , which is a map L n K , ¯0 ( V , . . . , V n ; V ) → L n K ( V , . . . , V n ; V ) (26) for any tuple V , . . . , V n , V of K -super vector spaces, is an isomorphism of K -modules.Proof. Definition (25) assigns to every f ∈ L n K , ¯0 ( V , . . . , V n ; V ) a functormorphism f . We have to show that one can reconstruct a unique f froma given f ∈ L n K ( V , . . . , V n ; V ). Let a seqeuence λ i ⊗ v i ∈ V i (Λ) be given,1 ≤ i ≤ n and let j ≤ n of these v i be odd, assuming for simplicity thatthese are the first j . By Λ ¯0 -linearity we then have f Λ ( λ ⊗ v , . . . , λ n ⊗ v n ) = λ n · · · λ j +1 f Λ ( λ ⊗ v , . . . , λ j ⊗ v j , ⊗ v j +1 , . . . , ⊗ v n ) . (27) I am grateful to V. Molotkov for pointing out an error in my original proof of thisProposition and for sending me a correct version. η : Λ j → Λ which is defined by η ( θ k ) = λ k ,where θ k , 1 ≤ k ≤ j are the odd generators of Λ j . In order to prove thestatement of the Proposition it will now be enough to show that there existsa unique g ∈ L n K , ¯0 ( V , . . . , V n ; V )such that f Λ j ( θ ⊗ v , . . . , θ j ⊗ v j , ⊗ v j +1 , . . . , ⊗ v n ) = θ j · · · θ ⊗ g ( v , . . . , v n ) . (28)This is sufficient because the fact that f is a functor morphism implies thatwe have a commutative square V (Λ j ) × . . . × V n (Λ j ) V ( η ) × ... × V n ( η ) / / f Λ j (cid:15) (cid:15) V (Λ) × . . . × V n (Λ) f Λ (cid:15) (cid:15) V (Λ j ) V ( η ) / / V (Λ) (29)and this then entails that f Λ ( λ ⊗ v , . . . , λ n ⊗ v n ) = λ n · · · λ ⊗ g ( v , . . . , v n )for a unique even linear map g .To prove that a unique g as claimed in eq. (28) exists, we first observethat the most general expression that could appear on the right hand sideof eq. (28) reads f Λ j ( θ ⊗ v , . . . , θ j ⊗ v j , ⊗ v j +1 , . . . , ⊗ v n ) = θ j · · · θ ⊗ g ( v , . . . , v n )++ X m It has to be shown that the assignment f f is a bijection( L K ) ¯0 ( V ; V ′ ) → L K ( V ; V ′ ) (32)for any pair V, V ′ of K -super vector spaces. This is a special case of Prop. 3.1if one inserts there V = V and V = V ′ .Actually one gets even more, since Prop. 3.1 makes a statement aboutmultilinear maps on an arbitrary finite number of arguments. Corollary 3.3 (see also [13]) . The functor · : SMod K ( Sets ) → Mod K ( Sets Gr ) induces fully faithful functors · : SLie K ( Sets ) → Lie K ( Sets Gr ) (33) · : SAlg K ( Sets ) → Alg K ( Sets Gr ) (34) between the categories of K -super Lie algebras and ordinary K -Lie algebrasin Sets Gr , and between K -super algebras and ordinary K -algebras.Proof. Clearly, · maps a morphism of K -super algebras into a unique mor-phism of the corresponding K -algebras. We have to check that this is alsoa surjective assignment. Let f : ( A, µ ) → ( A ′ , µ ′ ) be a morphism of K -algebras, where µ, µ ′ are the multiplications. Then for every Λ ∈ Gr , wehave f Λ ( µ Λ ( λ ⊗ a , λ ⊗ a )) = µ ′ Λ ( f Λ ( λ ⊗ a ) , f ( λ ⊗ a )) . (35)for all λ i ⊗ a i ∈ A (Λ). The left hand side is f Λ ( λ λ ⊗ µ ( a , a )) = λ λ ⊗ f ( µ ( a , a )) , (36)while the right hand side can be written as µ ′ Λ ( λ ⊗ f ( a ) , λ ⊗ f ( a )) = λ λ ⊗ µ ′ ( f ( a ) , f ( a )) , (37)where f, µ, µ ′ are the maps of K -super vector spaces corresponding to thebarred versions by Prop. 3.1. Thus these maps are K -super algebra homo-morphisms.This argument applies to every K -multilinear superalgebraic structurewhich is defined by relations involving finitely many arguments, e.g., asso-ciative superalgebras [13]. 12 efinition 3.4. A K -module V in Sets Gr is called superrepresentable if itis isomorphic to V for some K -super vector space V . Due to Prop. 3.1, the superrepresentable K -modules form a full subcat-egory in Mod K ( Sets Gr ), and · is an equivalence between this subcategoryand SMod K ( Sets ). Non-superrepresentable K -modules do indeed exist. Anexample that will prove useful later on is V nil . Let V be some K -supervector space and let Λ nil be the nilpotent ideal in Λ. Then one can define a K -module by setting V nil (Λ) := (Λ nil ⊗ K V ) ¯0 (38) V ( ϕ ) := ( ϕ ⊗ id V ) (cid:12)(cid:12) V nil (Λ) for ϕ : Λ → Λ ′ . (39)For every Λ, one has V (Λ) = V ¯0 ⊕ V nil (Λ) = V ¯0 ⊕ (cid:16) Λ nil ¯0 ⊗ V ¯0 (cid:17) ⊕ (cid:16) Λ nil ¯1 ⊗ V ¯1 (cid:17) . (40)Hence V nil is superrepresentable if and only if V ¯0 = 0, in which case V itselfsuperrepresents it.Finally, the change of parity functor Π can be carried over to the categoryof superrepresentable K -modules in an obvious way. Definition 3.5. Let SRepMod K ⊂ SMod K be the full subcategory of super-representable K -modules. The change of parity functor Π is defined as Π : SRepMod K → SRepMod K (41) V (Π( V )) . (42)Prop. 3.1 also ensures the existence of inner Hom objects in the category SRepMod K . As the following Corollary shows, we may choose these to bejust the “barred” versions of the inner Hom objects in SMod K ( Sets ). Corollary 3.6. Let V , . . . , V n , V be superrepresentable K -modules. Thenthere exists a superrepresentable inner Hom object L n K ( V , . . . , V n ; V ) ∼ = Hom( V , . . . , V n ; V ) = L n K ( V , . . . , V n ; V ) . (43) Proof. If a superrepresentable inner Hom object exists, it has to satisfy L K ( W ; L n K ( V , . . . , V n ; V )) ∼ = L n +1 K ( W , V , . . . , V n ; V )for all superrepresentalble K -modules W . By Prop. 3.1, the space on theright hand side is isomorphic to L n +1 K , ¯0 ( W, V , . . . , V n ; V ) ∼ = L K , ¯0 ( W, Hom( V , . . . , V n ; V ))13ecause inner Hom objects exist in SVec K . This in turn is, by Prop. 3.1,isomorphic to L K ( W , Hom( V , . . . , V n ; V )) , so Hom( V , . . . , V n ; V ) = L n K ( V , . . . , V n ; V ) satisfies the conditions of aninner Hom object in SRepMod K .This means we may choose the internal Hom-functors L n K : ( Mod ◦ K ) n × Mod K → Mod K to be given by the prescription L n K ( V , . . . , V n ; V ) = L n K ( V , . . . , V n ; V )for all super vector spaces V , . . . , V n , V , and we will do so in the following.In a completely analogous way one shows that SRepMod K possesses ten-sor products over K , and that these tensor products inherit all the usualproperties from SVec K . Sets Gr and a criterion for superrepresentabil-ity Before displaying some examples for the use of the categorical formulation wewill construct a set of generators for Sets Gr . This set will turn out to generatemany of the subcategories of Sets Gr we are interested in, and moreover willprovide an interpretation of the point sets produced by the functor of pointsused in supergeometry.Consider the functors P (Λ) in Sets Gr defined by P (Λ) : Λ ′ Hom SAlg K (Λ , Λ ′ ) (44)( ϕ : Λ ′ → Λ ′′ ) (cid:18) P (Λ)( ϕ ) : P (Λ)(Λ ′ ) → P (Λ)(Λ ′′ ) u ϕ ◦ u (cid:19) . From Prop. 2.4, we see that P (Λ n ) ∼ = R | n . The functors P (Λ) are thus justthe superpoints defined in Section 2.2.1. Lemma 3.7. For any functor F ∈ Sets Gr one has a bijection F (Λ) ∼ =Hom( P (Λ) , F ) .Proof. This is just the statement of the Yoneda lemma for the catgory Gr ,which asserts that the mapHom Sets Gr (Hom(Λ , − ) , F ) → F (Λ) (45) η η Λ (id Λ ) , (46)where η Λ is the Λ-component of the functor morphism η , is a bijection.14e recall the following definition. Definition 3.8. Let C be a category. A set { G i ∈ C } i ∈ I of objects for someindex set I is called a set of generators for C if for any pair f, g : A → B of distinct morphisms between A, B ∈ C there exists i ∈ I and a morphism s : G i → A such that the compositions G i s / / A f / / g / / B (47) are still distinct. A set of generators is thus able to keep all distinct morphisms distinctunder precomposition. Clearly, given a functor F : C → Sets , it is enough toconsider all the G i -points of it in order to find out whether it is representable.Obviously, if a generator set exists, then it need not at all be unique. Corollary 3.9. The set {P (Λ) | Λ ∈ Gr } is a set of generators for Sets Gr .Proof. Let Φ , Ψ : F → G be two distinct morphisms in Sets Gr , i.e., thereexists a Λ ∈ Gr and x ∈ F (Λ) such that Φ Λ ( x ) = Ψ Λ ( x ). Then any morphism η : P (Λ) → F which maps η Λ (id Λ ) = x separates Φ and Ψ.Note, however, that although the P (Λ) are superrepresentable K -mod-ules, they do not generate the subcategory Mod K ( Sets Gr ) as K -modules (ofcourse, they generate it as a restriction of Sets Gr ). The reason is that theadditional structure of a K -module restricts the morphisms in such a waythat the superpoints are not sufficient anymore to reproduce the Λ-points.For example, for any super vector space V over K we haveHom Mod K ( Sets Gr ) ( P ( K ) , V ) ∼ = Hom SVec K ( K | , V ) = { } , by Prop. 3.1 while Hom Sets Gr ( P ( K ) , V ) ∼ = V ¯0 as sets.We can obtain a statement analogous to Lemma 3.7 by considering theGrassmann superalgebras as super vector spaces and forming their functorsΛ. Then we obviously haveHom(Λ , V ) ∼ = Hom SVec K (Λ , V ) ∼ = (Λ ¯0 ⊗ V ¯0 ) ⊕ (Λ ¯1 ⊗ V ¯1 ) ∼ = V (Λ) . (48)The following Lemma shows that there is a simpler possibility. Lemma 3.10. { K | } is a generator set for SVec K .Proof. It is well known that { K } is a generator set for the category of K -vector spaces. Two morphisms of super vector spaces may differ either inan even or an odd subspace, so one even and one odd dimension is sufficientto separate all distinct morphisms between super vector spaces.15rop. 3.1 then implies that K | is sufficient to generate SRepMod K ( Sets Gr ),by Hom SRepMod K ( K | , V ) ∼ = Hom SVec K ( K | , V ) ∼ = V ¯0 ⊕ V ¯1 . (49)The fact that Λ ∼ = R | is already a set of generators shows that super-representability is a severe restriction on a K -module. Proposition 3.11. A K -module V ∈ Sets Gr is superrepresentable if andonly if1. the terminal morphism V ( ǫ Λ ) : V (Λ ) → V ( K ) is surjective and2. one has V ∼ = V ( K ) ⊕ Π(ker V ( ǫ Λ )) . (50) Proof. That follows directly from the definition a superrepresentable moduleand from (23).For non-superrepresentable modules, K | is in general not a generatorset. As a small demonstration of the uses of the categorical approach, we willshow how the definition of the supertrace of an endomorphism of a K -supervector space V is induced from the braiding isomorphisms in SVec K .The endomorphisms End( V ) = Hom( V, V ) are, as remarked above, onlya K -vector space. In the spirit of the functor of points, we should ratherstudy the inner Hom-object End( V ) = Hom( V, V ). Suppose we were notable to determine this super vector space by direct manipulations. Then wecould construct it using { Λ } as a set of generators:Hom( V , V )(Λ) = Hom(Λ ⊗ V , V ) ∼ = (Λ ¯0 ⊗ Hom( V, V )) ⊕ (Λ ¯1 ⊗ Hom(Π( V ) , V )) . These are clearly the points of a superrepresentable module associated toEnd( V ) = Hom( V, V ) ⊕ Π(Hom(Π( V ) , V )) , which coincides, of course, with the common definition as the superspace ofeven and odd linear morphisms V → V . Its elements may be interpreted asmatrices whose diagonal V ¯0 × V ¯0 - and V ¯1 × V ¯1 -blocks contain even entries(i.e., are from K ) and the off-diagonal blocks contain odd entries (are fromΠ( K )). The sets Hom( V , V )(Λ) can be interpreted in the very same way,with the even blocks then containing elements of Λ ¯0 and the odd blockselements of Λ ¯1 .The dual super vector space V ∗ is by definition V ∗ := Hom( V, K ) , 16o it is V ∗ ( K ) ∼ = Hom SVec K ( V, K ) = V ∗ ¯0 V ∗ (Λ ) ∼ = Hom SVec K ( V ⊕ Π( V ) , K ) = V ∗ ¯0 ⊕ V ∗ ¯1 , where V ∗ ¯1 is the ordinary dual space, i.e., V ∗ ¯1 = Hom Vec K ( V ¯1 , K ). One easilychecks that the higher points are just those of a superrepresentable moduledefined by V ∗ = V ∗ ¯0 ⊕ Π( V ∗ ¯1 ) . Now on one hand, we have an isomorphism φ : V ⊗ V ∗ → End( V ) (51) v ⊗ f ( w f ( w ) · v ) . On the other hand, we have the evaluation mapev : V ∗ ⊗ V → K (52) f ⊗ v f ( v ) . (53)The trace is generally defined to be the compositionTr : End( V ) φ − −→ V ⊗ V ∗ c V,V ∗ −→ V ∗ ⊗ V ev −→ K (54)Now the minus sign occuring in the super trace is induced from the braidingisomorphism c V,V ∗ . Let { e i } be a basis of V and { e j } be the dual basisof V ∗ . Then for ψ ∈ End( V ), φ − ( ψ ) is simply its decomposition intothe basis, ψ ij e i ⊗ e j . Applying c V,V ∗ yields ( − p ( e i ) p ( e j ) ψ ij e j ⊗ e i and thus,Tr( ψ ) = ( − p ( e i ) ψ ii .This example shows that the categorical approach can be a guideline howto “superize” notions from algebra if guessing them is impossible. It is alsoclear that the direct use of odd and even elements can be more elegant andtransparent, if one deals with a finite-dimensional or linear problem. Butespecially in geometry, as will be discussed next, the categorical method cansometimes be the only viable one. We will now proceed to define superdomains and supermanifolds using thecategorical formalism introduced in the previous section. Superdomains willbe defined as open subfunctors of superrepresentable K -modules. Since wecan define K -super vector spaces and thus K -modules of arbitrary dimen-sions, infinite-dimensional superdomains will also become available. Theseopen subfunctors will be glued in a straightforward way to form (possiblyinfinite-dimensional) supermanifolds.17o make the notion of an open subobject sensible, one has to define ananalogue of a topology on K -modules. Since the category Mod K ( Sets Gr )is not concrete, we will have to resort to the more abstract concept of aGrothendieck topology (for background, see e.g., [6], [1], [11]). Top Gr To define supermanifolds, the only functors of Sets Gr which are of relevanceare superrepresentable K -modules. But these are not merely functors into Sets , but actually into the category Top of topological spaces if we assumethat the even and odd parts of the super vector spaces V that are used canbe given the structure of topological vector spaces. Definition 4.1. Let F , F ′ be functors in Top Gr . F ′ is called a subfunctorof F , if1. for every Λ ∈ Gr , F ′ (Λ) is a topological subspace of F (Λ) , and2. the family of inclusions {F ′ (Λ) ⊂ F (Λ) (cid:12)(cid:12) Λ ∈ Gr } forms a functor mor-phism.In this case, one just writes F ′ ⊂ F . F ′ is called an open subfunctor of F if, in addition, each F ′ (Λ) is open in F (Λ) . Definition 4.2. Let F ′ , F ′′ be subfunctors of F ∈ Top Gr . Then the inter-section F ′ ∩ F ′′ is the functor whose points are ( F ′ ∩ F ′′ )(Λ) := F ′ (Λ) ∩ F ′′ (Λ) . (55) The union F ′ ∪ F ′′ is the functor defined by ( F ′ ∪ F ′′ )(Λ) := F ′ (Λ) ∪ F ′′ (Λ) . (56) A morphism ϕ : Λ → Λ ′ is mapped by F ′ ∩ F ′′ resp. F ′ ∪ F ′′ to the corre-sponding restrictions F ( ϕ ) (cid:12)(cid:12) F ′ (Λ) ∩F ′′ (Λ) resp. F ( ϕ ) (cid:12)(cid:12) F ′ (Λ) ∪F ′′ (Λ) . Clearly, both F ′ ∩ F ′′ and F ′ ∪ F ′′ are again subfunctors of F . Thefunctor emp : Λ → ∅ is the initial object in Top Gr . Definition 4.3. A functor morphism g : F ′′ → F is called open if thereexists a factorization g : F ′′ f −−−−→ F ′ ⊂ F such that f is an isomorphism and F ′ is an open subfunctor of F . Finally, the notion of an open covering can be carried over straightfor-wardly. 18 efinition 4.4. A family { u α : U α → F } of open functor morphisms iscalled an open covering of F if for each Λ ∈ Gr , the family of maps u α, Λ : U α (Λ) → F (Λ) is an open covering of the topological space F (Λ) . It is obvious from the definitions that this definition of open coveringsendows Top Gr with a Grothendieck topology, namely by simply pulling backthe global classical topology from Top . We will from now on always assumethat Top Gr is endowed with this particular topology.An example of an open subfunctor can be constructed in the followingway: let F ∈ Top Gr be an arbitrary functor, and let U ⊂ F ( R ) be an opensubset of its underlying set. Then we can construct an open subfunctor U ⊂ F by setting U ( R ) := U U (Λ) := F ( ǫ Λ ) − ( U ) ⊂ F (Λ) (57) U ( ϕ ) := F ( ϕ ) (cid:12)(cid:12) U (Λ) for ϕ : Λ → Λ ′ . Here, ǫ Λ : Λ → R is the terminal morphism of Λ ∈ Gr . It is clear from thedefinition that the inclusion U ⊂ F is indeed a functor morphism. We willdenote subfunctors of this form by U = F (cid:12)(cid:12) U and call them restrictions. Itwill turn out that only such subfunctors qualify as “affine” domains for theconstruction of supermanifolds. Now everything is prepared for the introduction of (possibly infinite-dimen-sional) superdomains. Definition 4.5. Let V be a superrepresentable K -module in Top Gr . V willbe called a locally convex, resp. Fr´echet, resp. Banach K -module if for every Λ ∈ Gr , the topological vector space V (Λ) is locally convex, resp. Fr´echet,resp. Banach. Definition 4.6. An open subfunctor F of a locally convex (resp. Fr´echet,resp. Banach) K -module in Top Gr is called a real (or complex, whichever K is) locally convex (resp. Fr´echet, resp. Banach) superdomain. Definition 4.7. A functor F ∈ Top Gr is called locally isomorphic to real (orcomplex) locally convex (resp. Fr´echet, resp. Banach) superdomains if thereexists an open covering { u α : U α → F } of F such that each U α is a locallyconvex (resp. Fr´echet, resp. Banach) superdomain. Restrictions, as it turns out, are the only open subfunctors a superrep-resentable K -module has. 19 roposition 4.8. Any open subfunctor U ⊂ V of a Banach (resp. Fr´echet,resp. locally convex) K -module V is a restriction V (cid:12)(cid:12) U , where U = U ( K ) (cf.(57)).Proof. Clearly, one can write V = V (cid:12)(cid:12) V ¯0 (58)for a superrepresentable K -module V which is represented by V , since V (Λ) = V ( ǫ Λ ) − ( V ¯0 ) . (59)Let now U be an arbitrary open subfunctor of V and U ⊂ V ¯0 be its K -points.The inclusion U ⊂ V must be a functor morphism, therefore the diagram U (Λ) ⊂ −−−−→ V (Λ) y U ( ǫ Λ ) y V ( ǫ Λ ) U ⊂ −−−−→ V ( K ) = V ¯0 (60)has to commute for all Λ ∈ Gr . This enforces U (Λ) = V ( ǫ Λ ) − ( U ) (61)for all Λ. For any morphism ϕ : Λ → Λ ′ of Grassmann algebras, we alsohave U (Λ) ⊂ −−−−→ V (Λ) y U ( ϕ ) y V ( ϕ ) U (Λ ′ ) ⊂ −−−−→ V (Λ ′ ) , (62)which commutes again, because the inclusion is a functor morphism. So, U ( ϕ ) = V ( ϕ ) (cid:12)(cid:12) U (Λ) . (63)By the properties of unions and intersections of open subfunctors, weobtain the following Corollary. Corollary 4.9. Let F ∈ Top Gr be a functor which is locally isomorphic toBanach (resp. Fr´echet, resp. locally convex) superdomains. Then every opensubfunctor U ⊂ F is a restriction F (cid:12)(cid:12) U .Proof. Let { u α : U α → F } be an open cover of F by superdomains of theappropriate type and let U be an arbitrary open subfunctor of F . Then every20ntersection U ∩U α is a superdomain, i.e. is a restriction F (cid:12)(cid:12) U ∩ U α , where U, U α are the underlying open sets of the respective functors. Therefore, U (Λ) = [ α ( F ( ǫ Λ )) − ( U α ∩ U ) = ( F ( ǫ Λ )) − ( U )By the same argument as in Prop. 4.8 (functoriality of inclusions), the imagesof the morphisms of Gr under U must be the restrictions of those of F .For the rest of this section, we will focus exclusively on Banach super-domains because Banach spaces are analytically nicest. After the necessarymodifications, most of the constructions described below will carry over atleast to the tame Fr´echet case. Definition 4.10. Let V (cid:12)(cid:12) U and V ′ (cid:12)(cid:12) U ′ be two real Banach superdomains. Afunctor morphism f : V (cid:12)(cid:12) U → V ′ (cid:12)(cid:12) U ′ is called supersmooth if1. the map f Λ : V (cid:12)(cid:12) U (Λ) → V ′ (cid:12)(cid:12) U ′ (Λ) (64) is smooth for every Λ ∈ Gr , and2. for every u ∈ V (cid:12)(cid:12) U (Λ) , the derivative Df Λ ( u ) : V (Λ) → V ′ (Λ) (65) is Λ ¯0 -linear. The second condition is necessary and sufficient to turn the sets of dif-ferential morphisms( Df ) Λ : V (cid:12)(cid:12) U (Λ) × V (Λ) → V ′ (Λ) (66)( Df ) Λ ( u, v ) = ( Df Λ ( u ))( v ) (67)into a V (cid:12)(cid:12) U -family of R -linear morphisms V → V ′ . This, in turn, has to berequired to make the differentiable structures on the sets V (cid:12)(cid:12) U (Λ), V ′ (cid:12)(cid:12) U ′ (Λ)functorial with respect to Λ, i.e., compatible with morphisms ϕ : Λ → Λ ′ .Together with supersmooth morphisms, smooth Banach superdomainsform a category BSDom . Replacing “smooth” with “real analytic” in Def. 4.10leads to the definition of the category of real analytic superdomains.For complex analytic domains, there seem to be two different approachesat first. One can start with superdomains in Top Gr which are isomorphic toopen subfunctors of superrepresentable C -modules, and define morphismsto be complex superanalytic functor morphisms. On the other hand, onecould as well use the category Gr C of Grassmann algebras over C from thevery beginning on, studying only functors in Top Gr C and using morphismswhich are analytic in their complex coordinates. However, the two resulting21ategories are equivalent: the Λ-points of some superrepresentable C module V are ( V ⊗ R Λ) ¯0 . But( V ⊗ R Λ) ¯0 ∼ = ( V R ⊗ R C ⊗ R Λ) ¯0 ∼ = ( V R ⊗ R Λ C ) ¯0 , (68)where V R is the real super vector space underlying V . The notion of su-persmoothness will be identical for both cases: if we use real Grassmannalgebras but C -modules, we will require the component maps of a functormorphism between superdomains to be holomorphic, which will imply thatthe differentials are C -linear. If we use complex Grassmann algebras, Λ ¯0 -linearity of the differentials will imply C -linearity and thus holomorphicityof the component maps of morphisms between superdomains. We will usethe term “supersmooth” in this sense from now on, i.e., denoting holomor-phicity in the complex case. An explicit description of supersmooth morphisms and a means to determinethe sets Hom BSDom ( U , V ) of morphisms between Banach superdomains willbe of great value for the following considerations. Theorem 4.11 (see also [13]) . A morphism f : V (cid:12)(cid:12) U → V ′ (cid:12)(cid:12) U ′ of Banachsuperdomains is supersmooth if and only if there exists a smooth map f : U → U ′ and for all k ≥ smooth maps f k : U → Sym k ( V ¯1 ; V ′ ) such that for every component f Λ of f , one has f Λ ( u + v nil + v nil ) = X k,m =0 k ! m ! D k f m ( u ) Λ ( v nil , . . . , v nil | {z } k times , v nil , . . . , v nil | {z } m times ) . (69) The collection of maps { f k | k ≥ } will be denoted as f • and will be calleda skeleton of the morphism f . A skeleton, if it exists, is unique. Some explanations are in order. The argument v ∈ V (Λ) of f Λ has beendecomposed according to (40) into its underlying part u ∈ V ¯0 and the parts v nil and v nil , which contain even and odd elements of Λ, respectively. Thisdecomposition is unique, cf. (40). The k-th differential D k f m ( u ) is, for each u ∈ U , a symmetric element of L k + m ¯0 ( V ⊗ k ¯0 , V ⊗ m ¯1 ; V ′ ), i.e., an even linear mapwhich is supersymmetric within its first k and within its last m arguments.Then D k f m ( u ) Λ is the Λ-component of the associated K -multilinear functormorphism. This functor morphism is uniquely determined by Prop. 3.1.22 roof. Let f : V (cid:12)(cid:12) U → V ′ (cid:12)(cid:12) U ′ be a given supersmooth morphism betweentwo Banach superdomains. We have to show that f gives rise to a uniqueskeleton f • .Clearly, f K is a smooth map U → U ′ and therefore qualifies as the re-quired map f . Let f Λ be one of the higher components of f . By assumption,it is a smooth map between the Banach domains V (cid:12)(cid:12) U (Λ) and V ′ (cid:12)(cid:12) U ′ (Λ). Let v = u + v nil + v nil be a point of V (cid:12)(cid:12) U (Λ). We can use the Taylor expansionaround u to write f Λ ( u + v nil + v nil ) = X k =0 k ! D k f Λ ( u )( v nil + v nil , . . . , v nil + v nil | {z } k times ) , (70)where the differentials D k f Λ are U -families of symmetric K -linear maps V (Λ) k → V ′ (Λ). Using linearity and symmetry, we can rewrite this as f Λ ( u + v nil + v nil ) = X k =0 ,m ≤ k m !( k − m )! D k f Λ ( u )( v nil , . . . , v nil | {z } ( k − m ) times , v nil , . . . , v nil | {z } m times ) . (71)Collecting all terms which depend on m copies of v nil produces the sum X k =0 k ! m ! D k + m f Λ ( u )( v nil , . . . , v nil | {z } k times , v nil , . . . , v nil | {z } m times ) . (72)By definition, the differentials of the component maps of a supersmoothmorphism are Λ ¯0 -linear. Therefore, the maps f Λ ,m ( u ) := D m f Λ ( u ) (cid:12)(cid:12) V ¯1 (Λ) : U → Sym m ( V ¯1 (Λ); V ′ (Λ)) (73)form the Λ-components of a smooth U -family of K -linear morphisms f m ( u ) : V ⊗ m ¯1 → V ′ . By Prop. (3.1), f m corresponds to a unique family of K -linearsupersymmetric maps f m : V ⊗ k ¯1 → V ′ . Then D k f m is, in turn, a smooth U -family of maps V ⊗ k ¯0 × V ⊗ m ¯1 → V ′ , which are supersymmetric within thetwo groups of arguments, and to which we assign a unique U -family D k f m .By uniqueness, the Λ-components of D k f m are identical to the D k f Λ in eq.(71), which also implies that the sums in eqs. (70) and (71) are finite.Clearly, the family { f Λ } uniquely determines the family { f m } , which isa skeleton for f . Conversely, every collection of smooth maps f n : U → Sym n ( V ¯1 ; V ′ ) obviously is the skeleton of a supersmooth functor morphism f : V (cid:12)(cid:12) U → V ′ , whose components are defined by (69). One also verifies thatcompositions of skeletons are skeletons for the composed maps.As pointed out in [13], one could use this result to define Banach super-domains completely without the use of functors and Grassmann algebras,namely as pairs ( U ⊂ V ¯0 , V ¯1 ), where both V ¯0 , V ¯1 are ordinary Banach spacesand U is an open domain in V ¯0 , and the morphisms between these pairs areskeletons. This is a point of view that we will not pursue here, however.23 .4 Banach supermanifolds A superrepresentable K -module V all of whose points V (Λ) have been en-dowed with Banach space structures in a functorial manner is not just afunctor Gr → Top , but actually a functor Gr → Man , where Man is the cate-gory of smooth Banach manifolds. In order to study Banach supermanifolds,we will now restrict our attention to the category Man Gr . Definition 4.12 (see also [13]) . Let F be a functor in Man Gr . An opencovering A = { u α : U α → F } α ∈ I of F is called a supersmooth atlas on F if1. every U α is a Banach superdomain,2. for every pair α, β ∈ I , the fiber product U αβ = U α × F U β ∈ Man Gr (74) can be given the structure of a Banach superdomain such that the pro-jections Π α : U αβ → U α and Π β : U αβ → U β are supersmooth.The maps u α : U α → F are called charts on F . Definition 4.13 (see also [13]) . Two supersmooth atlases A , A ′ on the func-tor F ∈ Man Gr are said to be equivalent if their union A ∪ A ′ is again a su-persmooth atlas on F . A supermanifold M is a functor in Man Gr endowedwith an equivalence class of atlases. The second condition in Definition 4.12 needs some explanation. Onemight think at first that the fiber product of any two supersmooth superdo-mains is automatically again a superdomain. But we may only assume herethat F is a functor in Man Gr , and thus we can a priori only assume the fiberproduct to exist in Man Gr . The projection morphisms Π α , Π β are thereforeonly guaranteed to be functor morphisms in Man Gr . The second conditiontherefore requires the fiber product of U α , U β to exist in the subcategory BSDom ⊂ Man Gr . Since there really exist functor isomorphisms in Man Gr which are not supersmooth (compare Thm. 4.11), this is not automatic. Definition 4.14 (see also [13]) . Let M , M ′ be Banach supermanifolds. Afunctor morphism f : M → M ′ is called supersmooth if for each pair ofcharts u : U → M , u ′ : U ′ → M ′ , the pullback U × M ′ U ′ Π (cid:15) (cid:15) Π ′ / / U ′ u ′ (cid:15) (cid:15) U u / / M f / / M ′ (75) can be given the structure of a Banach superdomain such that its projections Π , Π ′ are supersmooth. 24t is clear that the composition of two supersmooth morphisms of Banachsupermanifolds is again supersmooth. Thus Banach supermanifolds form acategory BSMan . If nothing else is specified, we will from now on only write SMan for the category of Banach supermanifolds, since we restrict ourselvesto them in this work. The set of supersmooth morphisms f : M → M ′ willbe denoted as SC ∞ ( M , M ′ ) := Hom BSMan ( M , M ′ ).According to Prop. 4.8, every open submanifold of a Banach superman-ifold M is of the form U = M (cid:12)(cid:12) U , where U is an open submanifold of theunderlying manifold M = M ( K ). Clearly, the restriction of the superman-ifold structure of M to U induces on the latter a unique supermanifoldstructure which makes the inclusion U ⊂ M a supersmooth morphism. Remark 4.15. The above definition implies that M = M ( R ) is a smoothmanifold and every point set M (Λ) has the structure of a smooth vectorbundle over M , with projection given by the image M ( ǫ Λ ) of the terminalmorphism ǫ Λ : Λ → R : M ( ǫ Λ ) : M (Λ) → M ( R ) = M. (76) If V is the super vector space on which M is modeled, then the fiber of thisbundle is isomorphic to the kernel of the map V ( ǫ Λ ) : V (Λ) → V ¯0 , which isjust V nil (Λ) (cf. eq. (38)). As we will discuss below, the similarity of thispicture with a de Witt supermanifold is not an accident. BSDom and SMan The categories BSDom and SMan inherit a topology from Top Gr in a quitestraightforward way. We note first that we can pull back the Grothendiecktopology from Top Gr to Man Gr along the forgetful functor N : Man Gr → Top Gr . This is equivalent to saying that a family { u α : U α → U } of mor-phisms in Man Gr is an open covering of U if and only if it is an open coveringof U when the u α are considered as morphisms between objects in Top Gr .In exactly the same way BSDom and SMan are supposed to inherit theirtopology as a subcategories of Man Gr .Although we will not make use of it in this article, we want to includein this Section the proof of a useful fact that will become important, e.g., inthe construction of super vector bundles over supermanifolds. Proposition 4.16. The Grothendieck topology on SMan is subcanonical,i.e., every representable functor on SMan is a sheaf.Proof. Let M be a given supermanifold and let { u α : U α → U } be anopen covering in SMan . The representable functor Hom( − , M ) defines apresheaf on SMan for which we have to ensure that the gluing axiom holds.25et f α : U α → M be supersmooth morphisms such that for every fiberedproduct U α × U U β the condition π ∗ α f α = π ∗ β f β holds, where π α,β : U α × U U β → U α,β are the canonical projections. Then wehave to show that there exists a unique supersmooth morphism f : U → M such that u ∗ α f = f α ∀ α. (77)Given the f α we can pointwise, i.e. for every Λ, patch together the smoothmaps f α Λ to yield smooth maps f Λ : U (Λ) → M (Λ) which are functorial inΛ, i.e., they form a functor morphism f in Man Gr satisfying (77). It thereforeonly remains to be shown that this f is supersmooth. We choose a covering { v β : V β → M} of M . Then for all α, β we have commutative diagrams U α × M V β π β / / π α (cid:15) (cid:15) V βv β (cid:15) (cid:15) U α u α % % JJJJJJJJJJJ f α / / MU f > > }}}}}}}} (78)where the upper square commutes and all its arrows are supersmooth be-cause the f α are supersmooth and the lower triangle commutes by con-struction. This is precisely the condition for f to be supersmooth, cf.Def. 4.14. SMan Superpoints were introduced in section 2.2.1 as linear supermanifolds cor-responding to purely odd super vector spaces. In Prop. 2.8, it was shownthat the category SPoint is dual to the category Gr , and a duality was cho-sen, namely P : Λ Spec (Λ) = ( {∗} , Λ). This amounts to the choice P (Λ n ) = K | n for the Grassmann algebra on n generators over K . But asupermanifold is also a functor Gr → Man . So, P can be considered as abifunctor P : Gr ◦ × Gr → Man . (79) Proposition 4.17. There exists an isomorphism of bifunctors P ∼ = Hom Gr ( − , − ) . (80) Proof. The Λ m -points of P (Λ n ) are P (Λ n )(Λ m ) = (Λ m ⊗ K | n ) ¯0 ∼ = Λ m, ¯1 ⊗ K n ∼ = Hom Gr (Λ n , Λ m ) , (81)where the last isomorphism was proved in Prop. 2.4.26e will now show that the superpoints generate the category SMan ,which basically means that they play the role for supermanifolds which isplayed by the manifold Spec K = ( {∗} , K ) for ordinary manifolds over K . Proposition 4.18. For M ∈ SMan and every Λ ∈ Gr , one has a bijection M (Λ) ∼ = SC ∞ ( P (Λ) , M ) = Hom SMan ( P (Λ) , M ) . (82) Proof. According to Thm. 4.11, each supersmooth map f : P (Λ n ) → M can be expressed in terms of a skeleton f • : {∗} → Sym • ( R | n ; V ′ ), where V ′ is the super vector space, on which M is locally modeled. Now f = f K : {∗} → M = M ( K ) is a map of the one point set into the underlyingmanifold M of M . Therefore we have a bijection SC ∞ ( P (Λ n ) , M ) ∼ = M × Sym k ≥ ( R | n ; V ′ ) . But Sym k ( R | n ; V ′ ) = ∧ k ( R n , V ¯ k ) = Hom( ∧ k R n , V ¯ k ) ∼ = Λ ( k ) n ⊗ K V ¯ k , where ∧ k are the alternating K -linear maps on k arguments, ¯ k is to beunderstood mod 2 and Λ ( k ) n denotes the elements of degree k in Λ n . Summingup, we have found Sym k ≥ ( R | n ; V ′ ) ∼ = V ′ nil (Λ n )and therefore SC ∞ ( P (Λ n ) , M ) ∼ = M × V ′ nil (Λ n ) . (83)As pointed out in Remark 4.15, this set is in bijection with M (Λ).The bijection constructed here is clearly functorial both with respect to M as well as with respect to Λ. This ensures that the set { Spec Λ n = P (Λ n ) | n ∈ N } is a set of generators for SMan . This set is not minimal however,since any of its infinite subsets is a generator set as well . A minimal set ofgenerators does not exist for SMan . Supergroups are a topic where the functor of points has been used alreadyfrom the very beginning on. In fact, the desire to construct super analogs ofLie groups was one of the ideas that led to the introduction of supermanifolds[8]. By definition, a Lie supergroup G is a group object in the category of I thank V. Molotkov for pointing this out to me. m : G × G → G i : G → G (84) e : P ( R ) → G such that each set G (Λ) becomes a group with group law m Λ , inversion i Λ and unit e Λ . So, a supergroup is not at all just a set with some specialgroup structure, but rather a tower of groups indexed by Gr and related bythe maps which are induced by functoriality with respect to Gr . Luckily,for matrix supergroups the situation simplifies greatly because they are allsubgroups of general linear supergroups, and these have global coordinates.For any K -super vector space V , the superspace End( V ) is obviouslya K -superalgebra with multiplication the composition of morphisms. As-suming V to be finite-dimensional and to be given in some fixed format(i.e., with fixed parities of the basis vectors), we can express the elements ofEnd( V ) as super matrices (cf. Sect. 3.4) and the multiplication µ : End( V ) × End( V ) → End( V )is just matrix multiplication. The algebra End( V ) is not supercommutative,of course. Switching to the associated K -module Hom( V , V ) translates µ into a functor morphism µ : Hom( V , V ) × Hom( V , V ) → Hom( V , V ) , whose Λ-component acts on λ ⊗ f, λ ′ ⊗ g ∈ Hom( V , V )(Λ) as µ Λ ( λ ⊗ f, λ ′ ⊗ g ) = λ ′ λ ⊗ µ ( f, g )in accord with our general definition (cf. eq.(25)). Therefore, we can rep-resent the elements of Hom( V , V )(Λ) also as matrices, whose entries arenow elements of Λ of the appropriate parity. In particular, every element A ∈ Hom( V , V )(Λ) can be written as A = a + c , where a is not proportionalto any element of Λ, i.e., a = Hom( V , V )( ǫ Λ )( A )where ǫ Λ : Λ → K is the terminal morphism and c is nilpotent. But thisshows that A is invertible if and only if a is invertible, and its inverse is A − = a − ∞ X n =0 ( − n ( ca − ) n . (85)The sum terminates after finitely many terms, since Λ is finitely generated.Thus the supergroup GL ( V ) is a superdomain, namely GL ( V ) = Hom( V, V ) (cid:12)(cid:12) GL ( V ¯0 ) . (86)28he entries of the matrices representing the elements of Hom( V, V ) = End( V )are therefore suited to be used as coordinates of the groups GL ( V ) and theirsubgroups.The above results have, of course, been known for a long time and can beobtained without the categorical framework. But not all groups are that sim-ple. In addition, the above reasoning also implies that the automorphisms ofinfinite-dimensional super vector spaces can be extended to Lie supergroupsand that their supermanifold structure is given as a restriction of Hom( V, V )to GL ( V ¯0 ). Another nontrivial example is the diffeomorphism supergroup d SD ( M ) of a supermanifold M , which will be studied in a subsequent paper[15]. It is instructive to recover the standard ringed space version of a superman-ifold from the categorical construction. This will, in addition, allow us torelate the categorical and the Berezin-Leites approach to the Rogers-de Wittapproach. It will also offer a new way to understand superfunctions as actual“maps” from somewhere to somewhere, although these maps will be functormorphisms. The idea developed here was first described by Molotkov [13].One defines an R -superalgebra R in SMan by setting R (Λ) := Λ (87) R ( ϕ ) := ϕ for ϕ : Λ → Λ ′ . (88)The R -superalgebra structure is provided by the Λ ¯0 -superalgebra structureson each Λ. Note that up to now, we never defined super objects in any of ourcategories of superobjects. It was never necessary – in fact one of the greatadvantages of the categorical approach is that one can work with purelyeven objects. Every K -supermodule is an ordinary K -module in Sets Gr . Inthis sense R is “super super”. The functor R is still superrepresentable (andhence indeed an object of SMan ) as we will show now, but the superalgebrarepresenting it is non-supercommutative.As an R -module, we have R ∼ = R ⊕ Π( R ) ∼ = R | . (89)We want to find a superalgebra structure on R | which represents the oneon R . Denoting the standard basis of R | as { , θ } , we can write R (Λ) = Λ ¯0 ⊕ Λ ¯1 ∼ = (Λ ¯0 ⊗ ⊕ (Λ ¯1 ⊗ θ ) . (90)29et µ : R × R → R denote the multiplication in R and let µ denote thehypothetical multiplication in R | that we want to determine. Let λ , λ ∈ Λ ¯0 be given. We have µ Λ ( λ ⊗ , λ ⊗ 1) = λ λ ⊗ µ (1 , . (91)Multiplying λ , λ within R (Λ) yields λ λ , which gets identified with λ λ ⊗ R | (Λ). Thus we have to require µ (1 , 1) = 1. Likewise, for λ ∈ Λ ¯0 and λ ∈ Λ ¯1 we have µ Λ ( λ ⊗ , λ ⊗ θ ) = λ λ ⊗ µ (1 , θ ) . (92)This must coincide with λ λ ⊗ θ , and thus we must have µ (1 , θ ) = θ .Analogously we find µ ( θ, 1) = θ .For λ , λ ∈ Λ ¯1 , however, µ Λ ( λ ⊗ θ, λ ⊗ θ ) = λ λ ⊗ µ ( θ, θ ) (93)enforces µ ( θ, θ ) = − R | endowed with this multiplication will be denotedas C s . As an R -algebra it is isomorphic to C , but as an R -superalgebra, it isisomorphic to C with i declared odd. It is non-supercommutative, in partic-ular, every non-zero element is invertible (which makes it a kind of super ana-log of a skew field). Note that, although C s is non-supercommutative, the R -superalgebra R is supercommutative. Just like passing to their functorsof points turns supercommutative algebras into commutative R -algebras,the special non-supercommutativity of C s is weakened to supercommuta-tivity of its functor of points as an R -algebra. It would be an interestingquestion to study whether this kind of reasoning can be iterated to yieldsomething like “super super super” objects and whether these would pos-sess any geometric interpretation. Molotkov in [13] proposes a formalism toinvestigate such questions, but a conclusive answer has yet to be found.The reason why we introduced R is that we need a superalgebra in SMan in order to induce the structure of a superalgebra on certain sets of mor-phisms which we want to interpret as the superfunctions on a supermanifold M . Consider the set of functor morphisms SC ∞ ( M ) := SC ∞ ( M , R ) . (94)Since R is a supercommutative R -superalgebra, SC ∞ ( M ) is canonicallyequipped with the structure of a supercommutative SC ∞ ( M , R )-superal-gebra. Moreover, we can embed R ֒ → SC ∞ ( M ) as the constant functions M → R . More precisely, for any r ∈ R , we define a supersmooth morphism f r : M → R by setting( f r ) Λ ( m ) = r for all Λ ∈ Gr , m ∈ M (Λ) . (95)30ia this embedding, SC ∞ ( M ) becomes endowed with an R -superalgebrastructure. We will call SC ∞ ( M ) the algebra of superfunctions on M .To see that this construction indeed produces the algebras of superfunc-tions which are used in the Berezin-Leites approach, consider the linearsupermanifold V associated with a finite-dimensional super vector space V .By Thm. 4.11, any supersmooth functor morphism f : V → R is given by asmooth map f : V ¯0 → R and by smooth maps V ¯0 → Sym n ( V ¯1 , R | ) n ≥ . The latter maps are just smooth maps V ¯0 → ∧ n ( V ¯1 , R ) ∼ = ∧ n V ∗ ¯1 where V ¯1 is now considered as an ordinary R -vector space, V ∗ ¯1 is its dualspace, and a map has as parity the exterior degree. Altogether, the set ofthese morphisms is therefore isomorphic to the superalgebra C ∞ ( V ¯0 ) ⊗ R ∧ • ( V ¯1 ) . (96)The superalgebra SC ∞ ( V , R ) can then also be written in terms of coordinatemaps x , . . . , x n , θ , . . . , θ m for V , as one usually does.Now let M be a supermanifold and let M be its underlying topologicalspace (i.e., the topological space underlying the base manifold M ( R )). Thenwe can assign to every open set U ⊂ M the R -superalgebra SC ∞ ( M (cid:12)(cid:12) U ) andto every inclusion U ′ ⊂ U of open sets the obvious restriction map. Thisyields a presheaf on M which is easily seen to satisfy the gluing axiom, i.e.,it is actually a sheaf S ( M ). Any morphism f : M → M ′ of supermani-folds induces, via its associated map M → M ′ of the underlying spaces, amorphism of sheaves S ( f ) : S ( M ) → S ( M ′ ). Therefore the assignment S : M 7→ S ( M ) (97) f S ( f ) (98)defines a functor from the category of supermanifolds to the category oftopological spaces ringed by supercommutative superalgebras.Denote by FinSMan the category of finite-dimensional supermanifoldsdefined by the categorical construction and let the category of Berezin-Leitessupermanifolds be the supermanifolds obtained from the standard ringedspace construction described in Sect. 2.2. Theorem 5.1 (see also [13]) . The functor S establishes an equivalence be-tween the category FinSMan and the category of Berezin-Leites supermani-folds. roof. The proof works exactly as in the non-super case. Given a Berezin-Leites supermanifold M = ( M, O M ) of dimension m | n , its sheaf is definedby the property that every of its stalks O M ,x for x ∈ M is isomorphic to asuperalgebra O M ,x ∼ = O K m ,p ⊗ ∧ • [ θ , . . . , θ n ]with odd quantities θ j , O K m being the sheaf of smooth or holomorphicfunctions and p ∈ K m . It is therefore isomorphic to a stalk of the sheaf S ( V ) for some super vector space V of dimension m | n . In finite dimensions,i.e., if the stalks are finitely generated, this isomorphism of stalks can alwaysbe extended to a local isomorphism of the corresponding sheaves, becausea sheaf is defined as a local homeomorphism onto its base space. Thisconstruction produces then a supermanifold defined in terms of charts bysuperdomains in V , so the functor S has a quasi-inverse.As is the case in ordinary geometry, this equivalence in general failsto hold if the stalks are not finitely generated, so in particular in infinitedimensions. In the de Witt approach [4], one works over a fixed Grassmann algebra Λwhich is either finitely generated or the direct limit Λ ∞ of the inclusionsΛ n − ֒ → Λ n , θ i → θ i , ≤ i ≤ n − . This algebra is then often called the “supernumbers”, and it indeedplays the role of numbers in this approach: superfunctions are thought ofas functions taking values in Λ, and superdomains of dimension m | n areconstructed as open domains in(Λ ¯0 ) m × (Λ ¯1 ) n . “Open” refers in this context to the de-Witt topology: a subset U ⊂ (Λ ¯0 ) m × (Λ ¯1 ) n is called open in this topology if and only if it has the form U = U × (Λ nil ¯0 ) m × (Λ ¯1 ) n , where U ⊂ K m is an open set in the ordinary sense.For finitely generated Λ, super geometry in the de Witt sense coincideswith the geometry of the set of Λ-points of a functor in Sets Gr . An opendomain of dimension m | n in the de Witt topology is isomorphic to the setof Λ-points of an open subfunctor U ⊂ V , where V is a superrepresentable K -module. Moreover, the superfunctions on a de Witt supermanifold arejust the Λ-components f Λ of functor morphisms f : M → R , as seen in theprevious subsection. 32o work with a fixed Λ is alright as long as one keeps all constructionsfunctorial under base change . This means that when one carries out someconstruction first for some Grassmann algebra Λ and then for Λ ′ , then anymorphism ϕ : Λ → Λ ′ must transform the first construction into the sec-ond. This excludes, in particular, any attempts to give the odd dimensionsa different topology than the trivial one (cf. Prop. 4.8). An example for theattempt to avoid artifical side effects from the choice of Λ is the prescriptionthat if one has chosen Λ = Λ n (i.e., the algebra on n generators), the coeffi-cient functions of any superfunctions must be restricted to Λ n − in order toavoid pathologies in the definition of partial derivatives [14]. Constructionsof this sort are completely unnecessary if one sticks to the more naturalrequirement of invariance under base change.The key advantage of the de Witt approach over the ringed space andthe categorical approach is its concreteness. Since the set of Λ-points of asupermanifold form an actual manifold and R (Λ) = Λ is just a superalgebra,one can preserve much of the intuitive formalism of ordinary differentialgeometry. Superfunctions are still maps from somewhere to somewhere,and domains are open sets of points in some topology. In most, but notall, practical situations it is sufficient to work with one set of Λ-points bychoosing a “generic” one, which usually just means that Λ has to be bigenough. Indeed, the vast majority of applications, especially in physics, hasbeen obtained that way.One just has to keep in mind that both from a mathematical but alsofrom a physical point of view, the choice of a fixed Λ can only be of anauxiliary nature. On the side of physics, the choice of a fixed Λ and the useof geometric constructions depending explicitly on it would introduce Λ asa fundamental ingredient of the theories constructed with it. There is noreason to believe that some particular Grassmann algebra Λ plays a specialrole in nature by providing the odd parameters appearing in supersymmetricfield theories. Moreover, the group Aut(Λ) would assume the illustrous roleof a group of invariances for all physical theories built this way. The onlypiece of information about an odd coordinate in a superspace that one isreally allowed to fix is that it is odd — anything beyond that is an arbitrarychoice and introduces additional “fake information”. References [1] M. Artin, A. Grothendieck, and M. Verdier. Th´eorie des topos etcohomologie ´etale des sch´emas, 1,2,3 , volume 270,305,569 of LectureNotes in Mathematics . Springer-Verlag, Berlin, 1963-1964. S´eminairede G´eom´etrie Alg´ebrique du Bois-Marie (SGA4).[2] F. A. Berezin and D. 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