aa r X i v : . [ m a t h . K T ] J un HOMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES
ENXIN WU
Abstract.
Diffeological spaces are natural generalizations of smooth manifolds, introduced byJ.M. Souriau and his mathematical group in the 1980’s. Diffeological vector spaces (especiallyfine diffeological vector spaces) were first used by P. Iglesias-Zemmour to model some infinitedimensional spaces in [I1, I2]. K. Costello and O. Gwilliam developed homological algebra fordifferentiable diffeological vector spaces in Appendix A of their book [CG]. In this paper, wepresent homological algebra of general diffeological vector spaces via the projective objects withrespect to all linear subductions, together with some applications in analysis.
Contents
1. Introduction 12. Definition and basic examples 33. Categorical properties 44. More examples 85. Fine diffeological vector spaces 106. Projective diffeological vector spaces 137. Homological algebra for diffeological vector spaces 16References 231.
Introduction
The concept of a diffeological space and the terminology was formulated by J.M. Souriau and hismathematical group in the 1980’s ([So1, So2]) as follows:
Definition 1.1. A diffeological space is a set X together with a specified set D X of functions U → X (called plots ) for every open set U in R n and for each n ∈ N , such that for all open subsets U ⊆ R n and V ⊆ R m : (1) (Covering) Every constant map U → X is a plot; (2) (Smooth compatibility) If U → X is a plot and V → U is smooth, then the composition V → U → X is also a plot; (3) (Sheaf condition) If U = ∪ i U i is an open cover and U → X is a set map such that eachrestriction U i → X is a plot, then U → X is a plot.We usually use the underlying set X to represent the diffeological space ( X, D X ) . Comparing with the concept of a smooth manifold, a diffeological space starts with a set insteadof a topological space, and it uses all open subsets of Euclidean spaces for characterizing smoothness,subject to the above three axioms. It is a sheaf over a certain site and has an underlying set. Thismakes it a concrete sheaf; see [BH].The theory of diffeological spaces was further developed by several mathematicians, especiallySouriau’s students P. Donato and P. Iglesias-Zemmour. Recently, P. Iglesias-Zemmour published
Date : July 31, 2018.2010
Mathematics Subject Classification. the book [I2]. We refer the reader unfamiliar with diffeological spaces to [I2] for terminology anddetails. Let us mention a few basic properties:A smooth map between diffeological spaces is a function sending each plot of the domain to aplot of the codomain. Diffeological spaces with smooth maps form a category, denoted by D iff. Thecategory D iff contains the category of smooth manifolds and smooth maps as a full subcategory, thatis, every smooth manifold is automatically a diffeological space, and smooth maps between smoothmanifolds in this new sense are the same as smooth maps between smooth manifolds in the usualsense. Moreover, the category D iff is complete, cocomplete and (locally) cartesian closed. Every(co)limit in D iff has the corresponding (co)limit in S et as the underlying set. For any diffeologicalspaces X and Y , we write C ∞ ( X, Y ) for the set of all smooth maps X → Y . There is a naturaldiffeology (called the functional diffeology ) on C ∞ ( X, Y ) making it a diffeological space. Also,every diffeological space has a natural topology called the D -topology , which is the same as theusual topology for every smooth manifold. See [I2, Chapters 1 and 2] for more details.Vector spaces are fundamental objects in mathematics. There are corresponding objects in diffe-ology called diffeological vector spaces, that is, they are both diffeological spaces and vector spacessuch that addition and scalar multiplication maps are both smooth. P. Iglesias-Zemmour focused onfine diffeological vector spaces and used them to model some infinite dimensional spaces in [I1, I2].K. Costello and O. Gwilliam developed homological algebra for differentiable diffeological vectorspaces (that is, diffeological vector spaces with a presheaf of flat connections) in Appendix A oftheir book [CG]. In this paper, we study homological algebra of general diffeological vector spaces.Here are the main results: (1) The category of diffeological vector spaces and smooth linear maps iscomplete and cocomplete (Theorems 3.1 and 3.3). (2) For every diffeological space, there is a freediffeological vector space generated by it, together with a universal property (Proposition 3.5). Everydiffeological vector space is a quotient vector space of a free diffeological vector space (Corollary 3.13),but not every diffeological vector space is a free diffeological vector space generated by a diffeologi-cal space (Example 3.7). (3) We define tensor products and duals for diffeological vector spaces inSection 3. (4) We define short exact sequences of diffeological vector spaces (Definition 3.15), andwe have necessary and sufficient conditions for when a short exact sequence of diffeological vectorspaces splits (Theorem 3.16). Unlike the case of vector spaces, not every short exact sequence ofdiffeological vector spaces splits (Example 4.3). We also get a generalized version of Borel’s theorem(Remark 4.5). (5) Fine diffeological vector spaces (Definition 5.2) behave like vector spaces, exceptfor taking infinite products and duals (Examples 5.4 and 5.5). Every fine diffeological vector spaceis a free diffeological vector space generated by a discrete diffeological space (in the list of factsabout fine diffeological vector spaces in Section 5). A free diffeological vector space generated by adiffeological space is fine if and only if this diffeological space is discrete (Theorem 5.3). (6) Everyfine diffeological vector space is Fr¨olicher (Proposition 5.7). (7) We define projective diffeologicalvector spaces as projective objects in the category of diffeological vector spaces with respect to linearsubductions (Definition 6.1), and we have many equivalent characterizations (Remark 6.9, Proposi-tion 6.14, Corollaries 6.13 and 7.9). Every fine diffeological vector space and every free diffeologicalvector space generated by a smooth manifold is projective (Corollaries 6.3 and 6.4). But not every(free) diffeological vector space is projective (Example 6.8). However, there are enough projectivesin the category of diffeological vector spaces (Theorem 6.11). (8) We have many interesting (non)-examples of (fine or projective) diffeological vector spaces (Examples 5.4, 5.5, 5.6, 6.6, 6.10, 6.8,Proposition 6.2, Corollaries 6.3 and 6.4, and Remark 6.7). (9) We establish homological algebrafor general diffeological vector spaces in Section 7. As usual, every diffeological vector space has adiffeological projective resolution, and any two diffeological projective resolutions are diffeologicallychain homotopic (see the paragraph before Lemma 7.1). Schanuel’s lemma, Horseshoe lemma and(Short) Five lemma still hold in this setting (Lemmas 7.1, 7.2 and 7.10, and Proposition 7.11). Forany diffeological vector spaces V and W , we can define ext diffeological vector spaces Ext n ( V, W )(Definition 7.4). For any short exact sequence of diffeological vector spaces in the first or secondvariable, we get a sequence of ext diffeological vector spaces which is exact in the category V ect of OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 3 vector spaces and linear maps (Theorems 7.6 and 7.7). And Ext ( V, W ) classifies all short exactsequences in D Vect of the form 0 → W → A → V → Convention : Throughout this paper, unless otherwise specified, • every vector space is over the field R of real numbers; • every linear map is over R ; • every tensor product is over R , i.e., ⊗ = ⊗ R ; • every smooth manifold is assumed to be Hausdorff, second-countable, finite dimensional andwithout boundary; • every smooth manifold is equipped with the standard diffeology; • every subset of a diffeological space is equipped with the sub-diffeology; • every product of diffeological spaces is equipped with the product diffeology; • every quotient set of a diffeological space is equipped with the quotient diffeology; • every function space of diffeological spaces is equipped with the functional diffeology.2. Definition and basic examples
In this section, we recall the definitions of diffeological vector spaces and smooth linear mapsbetween them, together with some basic examples.
Definition 2.1. A diffeological vector space is a vector space V together with a diffeology, suchthat the addition map V × V → V and the scalar multiplication map R × V → V are both smooth. Definition 2.2. A smooth linear map between two diffeological vector spaces is a function whichis both smooth and linear. Diffeological vector spaces with smooth linear maps form a category, denoted by D Vect. Giventwo diffeological vector spaces V and W , we always write L ∞ ( V, W ) for the set of all smooth linearmaps V → W . Example . Every vector space with the indiscrete diffeology is a diffeological vector space. Wewrite R ind for the diffeological vector space with the underlying set R and the indiscrete diffeology. Example . Every linear subspace of a diffeological vector space is a diffeological vector space.
Example . Every product of diffeological vector spaces is a diffeological vector space. In particular, Q ω R , the product of countably many copies of R , is a diffeological vector space. Example . Every quotient vector space of a diffeological vector space is a diffeological vectorspace.
Example . Let X be a diffeological space, and let V be a diffeological vector space. Then C ∞ ( X, V ) with pointwise addition and pointwise scalar multiplication is a diffeological vector space.Moreover, the map m : C ∞ ( X, R ) × C ∞ ( X, V ) → C ∞ ( X, V ) with m ( f, g )( x ) = f ( x ) g ( x )is a well-defined smooth map. Example . Every topological vector space with the continuous diffeology is a diffeological vectorspace. Moreover, if we write T Vect for the category of topological vector spaces and continuouslinear maps, then the adjoint pair (see [CSW, Proposition 3.3]) D : D iff ⇋ T op : C induces an adjoint pair D : D Vect ⇋ T Vect : C. ENXIN WU Categorical properties
In this section, we study the categorical properties of the category D Vect of diffeological vectorspaces and smooth linear maps. All categorical terminology is from [M]. We begin by showingthat D Vect is complete and cocomplete. Then we focus on the algebraic aspects of diffeologicalvector spaces. We construct the free diffeological vector space over an arbitrary diffeological space,and define tensor products, duals and short exact sequences of diffeological vector spaces. We alsodiscuss some isomorphism theorems and necessary and sufficient conditions for when a short exactsequence of diffeological vector spaces splits. Unlike the case of vector spaces, we will see in nextsection that not every short exact sequence of diffeological vector spaces splits.
Theorem 3.1.
The category D Vect is complete.Proof.
This follows directly from Example 2.4 and Example 2.5. (cid:3)
The underlying set of coproduct in D iff is disjoint union of the underlying sets, so coproduct in D iff cannot be coproduct in D Vect. On the other hand, the forgetful functor D Vect → V ect is aleft adjoint, so if coproduct exists in D Vect, the underlying set must be direct sum of the underlyingvector spaces. The proof of the following proposition tells us how to put a suitable diffeology on thisset to make it a coproduct in D Vect.
Proposition 3.2.
The category D Vect has arbitrary coproducts.Proof.
Let { V i } i ∈ I be a set of diffeological vector spaces. If the index set I is finite, then one can seedirectly that coproduct of these V i exists in D Vect, and it is Q i ∈ I V i with the product diffeology. Fora general index set I , we first construct a category I with objects finite subsets of I and morphismsinclusion maps. It is clear that the category I is filtered. There is a canonical functor G : I → D Vectsending J ⊆ J ′ to the natural map Q j ∈ J V j → Q j ∈ J ′ V j . Write U : D Vect → D iff for the forgetfulfunctor. Then by the filteredness of I , it is easy to show that colim( U ◦ G ) in D iff is a diffeologicalvector space. Hence, it is coproduct of { V i } i ∈ I in D Vect. In other words, it is the set V = ⊕ i ∈ I V i with the final diffeology for all canonical maps Q j ∈ J V j → V for all finite subsets J of the index set I . (cid:3) Theorem 3.3.
The category D Vect is cocomplete.Proof.
This follows directly from Example 2.6 and Proposition 3.2. (cid:3)
Now we discuss how to define hom-objects in D Vect:Recall that given two diffeological vector spaces V and W , we write L ∞ ( V, W ) for the set of allsmooth linear maps V → W . Since L ∞ ( V, W ) is a linear subspace of C ∞ ( V, W ), by Example 2.4and Example 2.7, L ∞ ( V, W ) with the sub-diffeology of C ∞ ( V, W ) is a diffeological vector space.From now on, L ∞ ( V, W ) is always equipped with this diffeology (called the functional diffeology )when viewed as a diffeological (vector) space. As an easy consequence, the evaluation map V × L ∞ ( V, W ) → W is smooth. Example . Let V be a diffeological vector space. Then the map L ∞ ( R , V ) → V defined by f f (1) is an isomorphism in D Vect.In order to define tensor products for diffeological vector spaces, we need the following proposition,which will be very useful throughout this paper.
Proposition 3.5.
The forgetful functor D Vect → D iff has a left adjoint.Proof. Given a diffeological space X , write F ( X ) for the free vector space generated by the under-lying set of X . For p : U → X, · · · , p n : U n → X plots of X , write R × U × · · · × R × U n → F ( X ) OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 5 for the map defined by ( r , u , . . . , r n , u n ) r [ p ( u )] + . . . + r n [ p n ( u n )]. Equip F ( X ) with thediffeology generated by all such maps for all n ∈ Z + . It is clear that this is the smallest diffeologyon F ( X ) such that F ( X ) is a diffeological vector space and the canonical map i X : X → F ( X ) issmooth. We call F ( X ) with this diffeology the free diffeological vector space generated by thediffeological space X . Moreover, we have the universal property that for any diffeological vectorspace V and any smooth map f : X → V , there exists a unique smooth linear map g : F ( X ) → V making the following triangle commutative: X f (cid:23) (cid:23) ✵✵✵✵✵✵ i X / / F ( X ) g (cid:3) (cid:3) ✟✟✟✟✟✟ V. Therefore, we can define F : D iff → D Vect by sending X → Y to the unique smooth linear map F ( X ) → F ( Y ) defined by the universal property. Then F is a functor which is left adjoint to theforgetful functor D Vect → D iff. (cid:3) In particular, if f : X → Y is a subduction of diffeological spaces, then F ( f ) : F ( X ) → F ( Y ) isa linear subduction of (free) diffeological vector spaces. Remark . The universal property of free diffeological vector spaces says that if X is a diffeologicalspace and V is a diffeological vector space, then there is a natural bijection between C ∞ ( X, V ) and L ∞ ( F ( X ) , V ). Indeed this is an isomorphism in D Vect.Similarly, if { X i } i ∈ I is a set of diffeological spaces and { V j } j ∈ J ∪{ V } is a set of diffeological vectorspaces, then we have natural isomorphisms in D Vect between C ∞ ( ` i ∈ I X i , V ) and Q i ∈ I C ∞ ( X i , V ),between L ∞ ( V, Q j ∈ J V j ) and Q j ∈ J L ∞ ( V, V j ), and between L ∞ ( ⊕ j ∈ J V j , V ) and Q j ∈ J L ∞ ( V j , V ). Example . Not every diffeological vector space is a free diffeological vector space generated by adiffeological space. R ind is such an example.Now we discuss tensor products in D Vect:Let V and W be two diffeological vector spaces. Since the vector space V ⊗ W is a quotient vectorspace of the free diffeological vector space F ( V × W ) generated by the product space V × W , byExample 2.6, V ⊗ W is a diffeological vector space with the quotient diffeology. From now on, wealways equip V ⊗ W with this diffeology.As usual, we have the following adjoint pair: Theorem 3.8.
For any diffeological vector space V , there is an adjoint pair − ⊗ V : D Vect ⇋ D Vect : L ∞ ( V, − ) . Proof.
This follows from the universal property of free diffeological vector space in the proof ofProposition 3.5, and the fact that for diffeological vector spaces V , W and A , a map W → L ∞ ( V, A )is smooth linear if and only if the adjoint set map W × V → A is smooth bilinear. (cid:3) Here are some basic properties of tensor product of diffeological vector spaces.
Remark . (1) Given diffeological vector spaces V , V and V , V ⊗ V is naturally isomorphic to V ⊗ V in D Vect, and ( V ⊗ V ) ⊗ V is naturally isomorphic to V ⊗ ( V ⊗ V ) in D Vect. The secondisomorphism follows from the fact that the canonical projection map F ( V × V × V ) toeither ( V ⊗ V ) ⊗ V or V ⊗ ( V ⊗ V ) is a (linear) subduction.(2) Given a set of diffeological vector spaces { V i } i ∈ I ∪ { W } , ( ⊕ i ∈ I V i ) ⊗ W is isomorphic to ⊕ i ∈ I ( V i ⊗ W ) in D Vect.(3) For any diffeological vector space V , the map V → V ⊗ R defined by v v ⊗ D Vect.
ENXIN WU
Proposition 3.10.
Let X and Y be diffeological spaces. Then F ( X × Y ) is isomorphic to F ( X ) ⊗ F ( Y ) in D Vect .Proof.
Since i X : X → F ( X ) and i Y : Y → F ( Y ) are smooth, so is i X × i Y : X × Y → F ( X ) × F ( Y )and hence the composite i F ( X ) × F ( Y ) ◦ ( i X × i Y ) : X × Y → F ( F ( X ) × F ( Y )), which induces a smoothlinear map F ( X × Y ) → F ( F ( X ) × F ( Y )). So we get a smooth linear map F ( X × Y ) → F ( X ) ⊗ F ( Y )given by P c j [ x j , y j ] P c j [ x j ] ⊗ [ y j ]. It is known from general algebra that this map is anisomorphism in V ect. We are left to show that the canonical projection map F ( F ( X ) × F ( Y )) → F ( X × Y ) given by P i a i [ P j b ij [ x j ] , P k c ik [ y k ]] P i,j,k a i b ij c ik [ x j , y k ] is smooth. This followsdirectly from the description of the diffeology on the free diffeological vector space generated by adiffeological space in the proof of Proposition 3.5. (cid:3) Now we discuss the dual to a diffeological vector space:Let V be a diffeological vector space. Write D ( V ) for the dual diffeological vector space L ∞ ( V, R ).Then D is a functor D Vect → D Vect op , and we have a natural transformation 1 → D : D Vect → D Vect.
Example . It is not true that for every finite dimensional diffeological vector space V , thecanonical map V → D ( V ) is a diffeomorphism. And it is not true that for every diffeological vectorspace V , the canonical map V → D ( V ) is injective. For example, D ( R ind ) = R = D ( R ind ). Onthe other hand, the canonical map V → D ( V ) is injective if and only if D ( V ) separates points,that is, for any v = v ′ ∈ V , there exists l ∈ D ( V ) such that l ( v ) = l ( v ′ ).Here are some isomorphism theorems: Proposition 3.12. (1)
Let f : V → W be a linear subduction between diffeological vector spaces. Then ˜ f : V / ker( f ) → W defined by ˜ f ( v + ker( f )) = f ( v ) is an isomorphism in D Vect . (2) If A is a diffeological vector space, B is a linear subspace of A , and C is a linear subspaceof B , then ( A/C ) / ( B/C ) is isomorphic to A/B in D Vect .Proof.
This is easy. (cid:3)
Now the following result is clear:
Corollary 3.13.
Every diffeological vector space is isomorphic to a quotient vector space of a freediffeological vector space in D Vect .Proof.
Let V be a diffeological vector space. Then the smooth map 1 V : V → V induces a smoothlinear map η : F ( V ) → V such that η ◦ i V = 1 V , where i V : V → F ( V ) is the canonical map. Thisequality implies that η is a subduction. Therefore, V is isomorphic to F ( V ) / ker( η ) in D Vect. (cid:3)
Remark . If V is a diffeological vector space, then the canonical map i V : V → F ( V ) is smoothbut not linear. Therefore, the short exact sequence 0 → ker( η ) → F ( V ) → V → not split smoothly in general; see Example 6.8.When V is projective (see Definition 6.1), the above short exact sequence splits smoothly. Definition 3.15.
Let A , V and B be diffeological vector spaces. Let i : A → V and p : V → B besmooth linear maps. We say that / / A i / / V p / / B / / is a short exact sequence in D Vect , if (1) it is a short exact sequence in V ect ; (2) i is an induction and p is a subduction.We call i a linear induction and p a linear subduction . OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 7
One can show easily that if f : V → W is a linear subduction between diffeological vector spaces,then f ∗ : L ∞ ( W, A ) → L ∞ ( V, A ) is a linear induction for any diffeological vector space A . Theorem 3.16.
Let / / A i / / V p / / B / / be a short exact sequence in D Vect . The following are equivalent. (1)
There exists a smooth linear map r : V → A such that r ◦ i = id A . (2) There exists a smooth linear map q : B → V such that p ◦ q = id B . (3) The short exact sequence is isomorphic to / / A i / / A × B p / / B / / in D Vect , with identity maps on A and B . In particular, V is isomorphic to A × B in D Vect . In this case, we say that the short exact sequence0 / / A i / / V p / / B / / D Vect splits smoothly , and we call A (and B ) a smooth direct summand of V . We willshow in Example 4.3 that not every short exact sequence in D Vect splits smoothly, and in particularnot every linear subspace or quotient vector space of a diffeological vector space is a smooth directsummand.
Proof.
If we only consider the statements in V ect instead of D Vect, then this is a standard resultfrom algebra; see [H, Theorem IV.1.18] for instance. We are left to prove the smoothness of certainmaps.(3) ⇒ (1) and (3) ⇒ (2) are clear.(1) ⇒ (3): By [H, The Short Five Lemma 1.17], ( r, p ) : V → A × B is an isomorphism in V ect.Its inverse can be written as i + q : A × B → V for some linear map q : B → V . It is straightforwardto check that q ◦ p = id V − i ◦ r . Hence, q ◦ p is smooth. Then p being a subduction implies that q is smooth. So i + q is smooth. Therefore, ( r, p ) is an isomorphism in D Vect, which implies theisomorphism of the two short exact sequences in D Vect.(2) ⇒ (3) can be proved dually as (1) ⇒ (3). (cid:3) Now the following result is direct:
Corollary 3.17.
Let V and W be diffeological vector spaces. (1) Let i : V → W and r : W → V be smooth linear maps such that r ◦ i = id V . Then thereexists a diffeological vector space X such that W is isomorphic to V × X in D Vect . (2) Let p : W → V and q : V → W be smooth linear maps such that p ◦ q = id V . Then thereexists a diffeological vector space X such that W is isomorphic to V × X in D Vect .Remark . It is easy to see that the category D Vect is additive with kernels and cokernels.However, it is not abelian, since a morphism in D Vect is monic if and only if the underlying set mapis injective, but not necessarily an induction. Indeed, D Vect is a quasi-abelian category in the senseof [Sc, Definition 1.1.3] with strict epimorphisms the linear subductions and strict monomorphismsthe linear inductions.
Remark . M. Vincent discussed tensor products and duals of diffeological vector spaces in hismaster thesis [V, Chapter 2]. He also showed that a vector space with a pre-diffeology (that is, aset of functions from open subsets of Euclidean spaces to this vector space) has a smallest diffeologycontaining the original pre-diffeology which makes it a diffeological vector space. This is moregeneral than the construction of free diffeological vector space generated by a diffeological space inthis section.
ENXIN WU More examples
In this section, we present two more examples of diffeological vector spaces from analysis. Theseexamples were introduced in the framework of Fr¨olicher spaces in [KM], and we adapt the proofs tothe diffeological setting. At the end, we also get a generalized version of Borel’s theorem.
Example . Let D + be the set of all smooth functions f : R n × R > → R such that ∂ | m | f∂ ( x,t ) m has continuous limits as t →
0, for every m ∈ N n +1 . Then D + is a linear subspaceof C ∞ ( R n × R > , R ). Moreover, there is a smooth linear map E : D + → C ∞ ( R n × R , R )such that E ( f )( x, t ) = f ( x, t ) when t > Remark . (1) It is shown in [Se] that the map E in the above example is continuous if both D + and C ∞ ( R n × R , R ) are equipped with several topologies. However, it is straightforward to seethat E is actually smooth in the diffeological sense. In particular, E is continuous if both D + and C ∞ ( R n × R , R ) are equipped with the D -topology.(2) The inclusion map i : R n × R > ֒ → R n × R induces a smooth linear map i ∗ : C ∞ ( R n × R , R ) → C ∞ ( R n × R > , R ) . It is clear that Im( i ∗ ) ⊆ D + . By abuse of notation, we write i ∗ : C ∞ ( R n × R , R ) → D + .Then i ∗ ◦ E = id D + . Therefore, E is an induction. In particular, if we equip C ∞ ( R n × R , R )with the D -topology, then the D -topology on D + is the initial topology with respect tothe map E . Moreover, the D -topology on C ∞ ( R n × R , R ) is the weak topology [CSW,Corollary 4.10], which is the same as ‘the topology of uniform convergence of each derivativeon compact subsets of R n +1 ’ [Se].(3) Let F = { f ∈ C ∞ ( R n × R , R ) | i ∗ ( f ) = 0 ∈ C ∞ ( R n × R > , R ) } . Then F is a linear subspace of C ∞ ( R n × R , R ). Moreover, C ∞ ( R n × R , R ) is isomorphic to D + × F in D Vect guaranteed by Corollary 3.17(1), with the isomorphism given by C ∞ ( R n × R , R ) → D + × F, h ( i ∗ ( h ) , h − E ◦ i ∗ ( h ))and D + × F → C ∞ ( R n × R , R ) , ( g, f ) E ( g ) + f. Example . Let φ : C ∞ ( R , R ) → Y ω R with ( φ ( f )) n = f ( n ) (0) . Then φ is a smooth linear map with kernel K = { f ∈ C ∞ ( R , R ) | f ( n ) (0) = 0 for all n ∈ N } . By Borel’s theorem (a more general version will be proved in Claim 1 below), φ is surjective.Therefore, there is a smooth linear bijection¯ φ : C ∞ ( R , R ) /K → Y ω R . Claim 1 (Generalized Borel’s theorem): The map ¯ φ is an isomorphism in D Vect.We are left to show that φ is a subduction. Let F : U → Q ω R be a plot. For any x ∈ U , fix twoopen neighborhoods V and V ′ of x in U , such that V ⊂ ¯ V ⊂ V ′ with ¯ V compact. Fix h ∈ C ∞ ( R , R )such that h has compact support and h ( t ) = t in an open neighborhood of 0. Let µ = 2 max { k F ( x ) k | x ∈ V } , OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 9 and let µ n = 2 max { µ n − } ∪ { k D α F n ( x ) n ! ( h n ) ( j ) ( t ) k | x ∈ V, | α | + j ≤ n, t ∈ R } . Then ( µ n ) is an increasing sequence with µ n ≥ n + 2 for all n ∈ N . Now we define˜ G : V × R → R by ˜ G ( x, t ) = ∞ X m =0 F m ( x ) m ! ( h ( tµ m ) µ m ) m . For any t = 0, this is a finite sum of smooth functions, hence ˜ G is smooth there. For t near 0, onthe one hand, one can show by Weierstrass M-test that ∞ X m =0 D α F m ( x ) m ! ( ( h ( tµ m ) m ) ( j ) µ mm )is uniformly convergent for all x ∈ V , α ∈ N dim( U ) and j ∈ N , so ˜ G is also smooth at t = 0, andhence ˜ G is smooth on V × R , that is, the adjoint map G : V → C ∞ ( R , R )is smooth; on the other hand, ˜ G ( x, t ) ∼ ∞ X m =0 F m ( x ) m ! t m as t → , so ( φ ◦ G ) n ( x ) = ∂ n ˜ G∂t n ( x,
0) = F n ( x )for all x ∈ V and n ∈ N . Therefore, φ is a subduction. Claim 2 : The D -topology on Q ω R is the product topology.Observe that the D -topology contains the product topology by the definition of product diffeology.Let A be a subset of Q ω R such that it is not in the product topology. In other words, there exists a = ( a , a , . . . ) ∈ A such that no open neighborhood of a in the product topology is contained in A . Let ǫ i ∈ R > for i ∈ N , and let U i = i Y j =0 ( a j − ǫ i , a j + ǫ i ) × Y k ∈ N \{ , ,...,i } R . Clearly, U i is an open neighborhood of a in the product topology, so there exists b i = ( b i , b i , . . . ) ∈ U i \ A . We can choose ǫ i ’s so that b in → a n fast as i → ∞ for each n ∈ ω ; for definition of fastconvergence, see [KM, page 17]. By the Special Curve Lemma [KM, page 18], there exists a smoothmap c : R → Q ω R such that c (1 /n ) = b n and c (0) = a . Therefore, A is not D -open. Claim 3 : There is no smooth linear map q : Q ω R → C ∞ ( R , R ) such that φ ◦ q = id Q ω R .To prove this, it is enough to show that for every smooth linear map q : Y ω R → C ∞ ( R , R ) ,φ ◦ q is not injective. By [CSW, Corollary 4.10], the D -topology on C ∞ ( R , R ) coincides with the weaktopology, or by [CSW, Proposition 4.2], the D -topology on C ∞ ( R , R ) contains the compact-opentopology. Hence, A = { f ∈ C ∞ ( R , R ) | | f ( t ) | < t with | t | ≤ } is D -open in C ∞ ( R , R ), which implies that q − ( A ) is D -open in Q ω R by the smoothness of q . It isclear that ~ , , . . . ) ∈ q − ( A ) since the zero function is in A and q is linear, and in Claim 2 weproved that the D -topology on Q ω R coincides with the product topology. So there exists N ∈ Z + such that ~ ∈ B = { x ∈ Y ω R | x n < /N for all n ≤ N } ⊆ q − ( A ) . Therefore, for any x ∈ Q ω R with x n = 0 for all n ≤ N , for any k ∈ N , kx ∈ B . So by the linearityof q k ( q ( x )) = q ( kx ) ∈ q ( B ) ⊆ A, which implies q ( x )( t ) = 0 for all t with | t | ≤ A . Therefore,( φ ◦ q ( x )) n = ∂ n q ( x ) ∂t n (0) = 0 for all n ∈ N , that is, φ ◦ q is not injective. Conclusion : Combining Claims 1 and 3 with Theorem 3.16, we know that0 / / K / / C ∞ ( R , R ) φ / / Q ω R / / D Vect, but it does not split smoothly. In particular, neither K = { f ∈ C ∞ ( R , R ) | f ( n ) (0) = 0 for all n ∈ N } nor Q ω R is a smooth direct summand of C ∞ ( R , R ), and C ∞ ( R , R ) is not isomorphic to K × Q ω R in D Vect.
Remark . On the contrary, for any n ∈ N , K n = { f ∈ C ∞ ( R , R ) | f ( i ) (0) = 0 for all i ≤ n } is asmooth direct summand of C ∞ ( R , R ). Remark . Combining Claim 1 of the above example and Re-mark 6.5, we can restate the generalized Borel’s theorem as follows: For any U open subset of aEuclidean space and any set of smooth functions { f i : U → R } i ∈ N , there exists a smooth map d : U × R → R such that ∂ k d∂y k ( x,
0) = f k ( x ) for all k ∈ N and x ∈ U .5. Fine diffeological vector spaces
In this section, we recall the definition and some basic properties of fine diffeological vectorspaces. They behave like vector spaces, except for infinite products and duals. We also present someexamples and non-examples of fine diffeological vector spaces. The analysis proof of Example 5.6relates to an interesting problem in analysis. We prove that the free diffeological vector spacegenerated by a diffeological space X is fine if and only if X is discrete, and that every fine diffeologicalvector space is Fr¨olicher. Proposition 5.1 ([I2, 3.7]) . Given any vector space V , there exists a smallest diffeology on V making it a diffeological vector space. Definition 5.2 ([I2, 3.7]) . The smallest diffeology in Proposition 5.1 is called the fine diffeology .A vector space with the fine diffeology is called a fine diffeological vector space . Here are some facts about fine diffeological vector spaces. • This diffeology is generated by all (injective) linear maps R n → V for all n ∈ N ; see [I2,3.8]. (In particular, this implies that the diffeological dimension of a diffeological vectorspace is always greater or equal to its vector space dimension. (See [I2, Chapter 1 and 2]and [Wu, Section 1.8] for the definition and basic properties of the diffeological dimensionof a diffeological space.) The equality does not always hold. For example, the diffeologicaldimension of R ind is ∞ .) • The fine diffeology on R n is the standard diffeology; see [I2, Exercise 66 on page 71]. • There is an equivalence between the category of fine diffeological vector spaces with smoothlinear maps and the category V ect of vector spaces with linear maps; see [I2, 3.10]. • The forgetful functor D Vect → V ect has both a left adjoint and a right adjoint. The leftadjoint is given by sending a vector space to the same space with the fine diffeology, andthe right adjoint is given by sending a vector space to the same space with the indiscretediffeology. In particular, if V is a fine diffeological vector space and W is a diffeologicalvector space, then L ∞ ( V, W ) = L ( V, W ), that is, a smooth linear map V → W is the sameas a linear map V → W . OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 11 • Every linear subspace of a fine diffeological vector space is again a fine diffeological vectorspace. • Finite product of fine diffeological vector spaces is again a fine diffeological vector space.But in general this is not true for infinite product; see Example 5.4. • Every quotient vector space of a fine diffeological vector space is again a fine diffeologicalvector space. Any coproduct of fine diffeological vector spaces in D Vect is again a finediffeological vector space; see Proposition 3.2. Therefore, any colimit of fine diffeologicalvector spaces in D Vect is again a fine diffeological vector space. • Every fine diffeological vector space is a free diffeological vector space generated by any basiswith the discrete diffeology. The converse is also true; see Theorem 5.3. • Let V be a fine diffeological vector space, and let A be a basis of V . Then V is isomorphicto the coproduct of | A | -copies of R in D Vect.Let W be an arbitrary diffeological vector space. Then L ∞ ( V, W ) is isomorphic in D Vectto the product of A -copies of W . In particular, if V = R n and W is fine, then L ∞ ( V, W ) isalso fine. • Tensor product of finitely many fine diffeological vector spaces is again a fine diffeologicalvector space. This follows from Proposition 3.10 and the fact that finite product of discretediffeological spaces is again a discrete diffeological space. • Let V be a fine diffeological vector space. Then the canonical map V → D ( V ) is injective.Furthermore, if V is also finite dimensional, then D ( V ) is fine and this canonical map is anisomorphism in D Vect. But in general, D ( V ) may not be fine; see Example 5.5. • Let 0 → V → V → V → V isfine, then this short exact sequence splits smoothly. If V is fine, then both V and V arefine, and hence this short exact sequence splits smoothly as well.As an immediate consequence, it is not true that for every diffeological vector space V ,there is a linear subduction from a fine diffeological vector space to V . Theorem 5.3.
Let X be a diffeological space such that F ( X ) is a fine diffeological vector space.Then X is discrete.Proof. Assume that there exist a plot p : U → X and a point u ∈ U such that for every openneighborhood V of u in U , there exists v ∈ V such that p ( v ) = p ( u ). Consider the plot q : R × U → F ( X ) given by ( t, u ) t [ p ( u )]. Since F ( X ) is fine, q = l ◦ f for some n ∈ N , some openneighborhoods A of t = 0 in R and B of u in U , a smooth function f : A × B → R n and a linearmap l : R n → F ( X ). Without loss of generality, we may assume that l sends the canonical basisof R n to the canonical basis of F ( X ), i.e., for every i ∈ { , . . . . , n } , l ( e i ) = [ x i ] for some x i ∈ X .Then the equality q = l ◦ f simply implies that f can not be smooth. Therefore, X is a discretediffeological space. (cid:3) Example . Let λ be a fixed smooth bump function, that is, λ ∈ C ∞ ( R , R ) such that supp( λ ) ⊂ (0 ,
1) and Im( λ ) = [0 , R → Q ω R be defined by(Φ( t )) n = λ [ n ( n + 1)( t − n + 1 )] . Then Φ is smooth, and the image of any neighborhood of 0 ∈ R under Φ does not live in any finitedimensional linear subspace of Q ω R . Therefore, Q ω R is not fine.Another way to see this is, if Q ω R were fine, then the short exact sequence of diffeological vectorspaces in Example 4.3 would split smoothly.Furthermore, ⊕ ω R is a linear subspace of Q ω R . But by using the function Φ it is easy to seethat the sub-diffeology from Q ω R is different from the fine diffeology on ⊕ ω R .Let i : R → Q ω R be defined by x ( x, , , . . . ). One can check easily that i is a linear inductionsuch that p ◦ i = 1 R . In other words, although Q ω R is not a fine diffeological vector space, it hasa smooth direct summand of a fine diffeological vector space. Example . The dual of a fine diffeological vector space may not be fine. For example, if V = ⊕ ω R ,the coproduct of countably many R ’s in D Vect, then D ( V ) = Q ω R , the product of countably many R ’s in D Vect. By Example 5.4, we know that D ( V ) is not fine. Example . The diffeological vector space C ∞ ( R , R ) is not fine.Here is an analysis proof. Assume that the diffeological vector space C ∞ ( R , R ) is fine. Let f : R → C ∞ ( R , R ) be the plot with f ( x )( y ) = e xy . Then there exist an open neighborhood U of0 in R , an integer n ∈ N , a smooth map g : U → R n and a linear map h : R n → C ∞ ( R , R ) suchthat f | U = h ◦ g . In other words, for any ( x, y ) ∈ U × R , e xy = P nk =1 g k ( x ) h k ( y ) for some smoothfunctions g , . . . , g n , h , . . . , h n ∈ C ∞ ( R , R ).Now fix δ ∈ R > such that δ, δ, . . . , nδ ∈ U . The ( n + 1) × ( n + 1) matrix A := [ e ijδ ] ni,j =0 isequal to n X k =1 [ g k (0) , g k ( δ ) , . . . , g k ( nδ )] T [ h k (0) , h k ( δ ) , . . . , h k ( nδ )] , that is, the sum of n rank 1 matrices, hence singular. On the other hand, A is a Vandermondematrix, so its determinant is Y i,j =0 ,...,ni>j ( e iδ − e jδ ) = 0 . The contradiction implies that the diffeological vector space C ∞ ( R , R ) is not fine.Here is an algebraic proof. In Claim 1 of Example 4.3, we have shown that0 / / K / / C ∞ ( R , R ) φ / / Q ω R / / C ∞ ( R , R )is not fine. The first way is, if C ∞ ( R , R ) is fine, then so is Q ω R , but it is not by Example 5.4.The second way is, if C ∞ ( R , R ) is fine, then the above short exact sequence splits smoothly, whichcontradicts the conclusion of Example 4.3.Fr¨olicher spaces are another well-studied generalization of smooth manifolds. We refer the readerto [St] for definition of the category F r of Fr¨olicher spaces and smooth maps. There is an adjointpair F : D iff ⇋ F r : G ; see [St]. We say that a diffeological space X is Fr¨olicher if there exists aFr¨olicher space Y such that G ( Y ) = X . Proposition 5.7.
Every fine diffeological vector space is Fr¨olicher.Proof.
Let V be a fine diffeological vector space. Write F = C ∞ ( V, R ) and write C = { c : R → V | l ◦ c ∈ C ∞ ( R , R ) for all l ∈ F} . Then ( C , V, F ) is a Fr¨olicher space. We are left to show that for anyopen subset U of R n , for any set map f : U → V , if l ◦ f ∈ C ∞ ( U, R ) for all l ∈ F , then f is a plotof V .Note that if A is a basis of V , then V is isomorphic to ⊕ a ∈ A R in D Vect. Hence the projectionof V to any 1-dimensional linear subspace is in F . Therefore, it is enough to show that for any u ∈ U , there exists an open neighborhood U ′ of u , such that Im( f | U ′ ) is in a finite dimensionallinear subspace of V .Now assume that for every open neighborhood U ′ of u in U , Im( f | U ′ ) is not in a finite dimensionallinear subspace of V . Then there exists a sequence u n → u in U such that { f ( u n ) } n ∈ N is a linearly independent subset of V, if f ( u ) = 0;or { f ( u n ) } n ∈ N ∪ { f ( u ) } is a linearly independent subset of V, if f ( u ) = 0 . We can extend this linearly independent subset to a basis A of V . Then by the universal propertyof coproduct, g : V → R defined by the linear extension of the map ( f ( u n ) n ∈ N other elements in A OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 13 is in F . But clearly g ◦ f is not continuous, hence not smooth, contracting the assumption that l ◦ f ∈ C ∞ ( U, R ) for all l ∈ F . (cid:3) Projective diffeological vector spaces
In this section, we introduce a large class of diffeological vector spaces called projective diffeologicalvector spaces. They are projective with respect to linear subductions. Fine diffeological vector spacesand the free diffeological vector spaces generated by smooth manifolds are such examples. We haveseveral equivalent characterizations of projective diffeological vector spaces in this section and next.However, not every (free) diffeological vector space is projective. But there are enough projectivesin the category D Vect of diffeological vector spaces.
Definition 6.1.
We call a diffeological vector space V projective , if for every linear subduction f : W → W and every smooth linear map g : V → W , there exists a smooth linear map h : V → W such that g = f ◦ h . Proposition 6.2.
Let X be a diffeological space. Then the free diffeological vector space F ( X ) generated by X is projective if and only if for any linear subduction f : W → W and any smoothmap g : X → W , there exists a smooth map h : X → W making the following diagram commutative: X g (cid:15) (cid:15) h | | ③③③③③③③③ W f / / W . Proof.
This follows directly from definition of projective diffeological vector space and the universalproperty of free diffeological vector space generated by a diffeological space. (cid:3)
Corollary 6.3.
Every fine diffeological vector space is projective.Proof.
This follows directly from Proposition 6.2 together with the fact that every fine diffeologicalvector space is a free diffeological vector space generated by a discrete diffeological space. (cid:3)
Corollary 6.4.
The free diffeological vector space generated by a smooth manifold is projective.Proof.
Let M be a smooth manifold. For any linear subduction f : W → W and any smooth map g : M → W , by Proposition 6.2, we only need to construct a smooth map h : M → W such that g = f ◦ h .Since f : W → W is a subduction and M is a smooth manifold, we can find an atlas { U i } i ∈ I of M such that for each i , U i is diffeomorphic to a bounded open subset of R n with n = dim( M ), andthere exists a smooth map h i : U i → W making the following diagram commutative: U i h i ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ (cid:31) (cid:127) / / M g / / W W . f O O Let { ρ i } i ∈ I be a partition of unity subordinate to this covering { U i } i ∈ I of M . By the sheaf condition, ρ i h i : M → W defined by m ρ i ( m ) h i ( m ) is smooth for each i . And since W is a diffeologicalvector space, h : M → W defined by h ( m ) = P i ∈ I ρ i ( m ) h i ( m ) is smooth. It is easy to check that f ( h ( m )) = g ( m ) for all m ∈ M . (cid:3) By Theorem 5.3, it is clear that Corollary 6.4 provides a lot of projective but not fine diffeologicalvector spaces.
Remark . The above corollary implies that for any linear subduction W → W , every plot of W globally lifts to a plot of W . This is clearly not true for general subductions.Moreover, by [I2, 8.15], we know that every linear subduction π : W → W is a diffeologicalprincipal bundle; see [I2, Chapter 8]. The pullback in D iff along any plot p : U → W is globally trivial. And for any ( u, w ) ∈ U × W with p ( u ) = π ( w ), there exists a smooth map f : U → W such that p = π ◦ f and f ( u ) = w . Example . (1) Assume that R / / (cid:15) (cid:15) R i (cid:15) (cid:15) R i / / X is a pushout diagram in D iff. Then F ( X ) is projective.Here is the proof. Let f : W → W be a linear subduction, and let g : X → W be a smooth map. By Corollary 6.4, there exists smooth maps α, ¯ β : R → W such that f ◦ α = g ◦ i and f ◦ ¯ β = g ◦ i . Since W is a diffeological vector space, we can define β : R → W by β ( y ) = ¯ β ( y ) − ¯ β (0) + α (0). Then the map β is smooth, and the linearity of f implies that f ◦ β = g ◦ i . Now α (0) = β (0). So we have a smooth map h : X → W suchthat f ◦ h = g . The result then follows from Proposition 6.2.(2) Let X be the union of the coordinate axes in R with the sub-diffeology. Then F ( X ) isprojective.Here is the proof. Write i , i : R → X for the smooth maps defined by i ( x ) = ( x, i ( y ) = (0 , y ), and write i : X → R for the inclusion. Let f : W → W be a linearsubduction, and let g : X → W be a smooth map. Since W is a diffeological vector space,we can define G : R → W by G ( x, y ) = g ( i ( x )) + g ( i ( y )) − g ( i (0)). Then G is a smoothmap, and G ◦ i = g . By Corollary 6.4, there exists a smooth map H : R → W such that f ◦ H = G . Hence h := H ◦ i : X → W is a smooth map such that f ◦ h = g . Remark . More generally, use the terminology in [CW], we have the following: Let X and Y bediffeological spaces. If either X is cofibrant, or F ( Y ) is a projective diffeological vector space andthere is a cofibration X → Y , then F ( X ) is a projective diffeological vector space. This follows fromthe fact that every linear subduction is a trivial fibration.However, not every (free) diffeological vector space is projective: Example . (1) Not every diffeological vector space is projective. From Example 4.3, we know that Q ω R is such an example.(2) More surprisingly, not every free diffeological vector space is projective.Let T α be the 1-dimensional irrational torus of slope some irrational number α , and let π : R → T α be the quotient map. Then F ( π ) : F ( R ) → F ( T α ) is a linear subduction. Weclaim that F ( T α ) is not projective.We only need to show that there exists no smooth map h : T α → F ( R ) such that i T α = F ( π ) ◦ h , where i T α : T α → F ( T α ) is the canonical smooth map. Otherwise, f := h ◦ π ∈ C ∞ ( R , F ( R )) and F ( π )( f ( x )) = i T α ( π ( x )) for all x ∈ R . On the one hand, f ∈ C ∞ ( R , F ( R ))implies that there exists a connected open neighborhood A of 0 in R together with smoothmaps α i : A → R (viewed as scalars) and β i : A → R (viewed as base) with 1 ≤ i ≤ n forsome minimum n ∈ Z + , such that f ( x ) = P ni =1 α i ( x )[ β i ( x )] for all x ∈ A . Since ( Z + α Z ) ∩ A is dense in A , the map β i must be constant for each i . On the other hand, since π | A : A → T α is surjective, F ( π )( f ( x )) = i T α ( π ( x )) for all x ∈ A implies that some β i can not be constant.So we reach the contradiction. OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 15
As consequences of this example, • we get an independent proof that any 1-dimensional irrational torus T α is not cofibrant(see [CW, Example 4.27] for another proof); • the diffeological vector space F ( T α ) is not fine; • projective diffeological vector spaces are not preserved under arbitrary colimits in D Vect.By a similar argument, one can show that F ( T α ) := F ( F ( T α )), the free diffeologicalvector space generated by F ( T α ) is also not projective. More generally, none of F n ( T α ) := F ( F n − ( T α )) is projective for n ∈ Z + .(3) R ind is not projective, since there is no smooth linear map h : R ind → F ( R ind ) such that η ◦ h = 1 R ind , where η : F ( R ind ) → R ind is the canonical smooth linear map induced by1 R ind : R ind → R ind .Here are some basic properties for projective diffeological vector spaces: Remark . (1) A diffeological vector space V is projective if and only if every linear subduction W → V splits smoothly. This follows from definition of projective diffeological vector space and thefact that linear subductions are closed under pullbacks in D Vect.(2) Let { V i } i ∈ I be a set of diffeological vector spaces. Then each V i is projective if and only ifthe coproduct ⊕ i ∈ I V i in D Vect is projective.(3) Projective diffeological vector spaces are closed under retracts in D Vect, that is, if f : V → W and g : W → V are smooth linear maps between diffeological vector spaces such that g ◦ f = 1 V and W is projective, then V is also projective.(4) Let 0 → V → V → V → V is projective, then this short exact sequence splits smoothly. In particular, every projectivediffeological vector space is a smooth direct summand of a free diffeological vector space,but the converse is not true in general. Example . (1) Let C n be the cyclic subgroup of order n for the multiplicative group S . Let C n act on R by rotation, and write X for the quotient diffeological space with the quotient map π : R → X . Then F ( X ) is a projective diffeological vector space. Here is the proof. Define f : X → F ( R ) by x P n [¯ x ], where the sum is over all ¯ x ∈ π − ( x ). Clearly f is a smoothmap such that F ( X ) is a retract in D Vect of the projective diffeological vector space F ( R ).Hence, F ( X ) is also projective.(2) Similarly, let X n be the quotient diffeological space R n / {± } . Then F ( X n ) is a projectivediffeological vector space.Recall from Example 6.8(2) that the domain of the canonical linear subduction F ( T α ) → F ( T α )is not projective. So the proof of Corollary 3.13 does not provide us a functorial way to find aprojective diffeological vector space V together with a linear subduction V → F ( T α ). However,there is a linear subduction F ( π ) : F ( R ) → F ( T α ) whose domain is a projective diffeological vectorspace. Theorem 6.11.
The category D Vect has enough projectives, that is, for any diffeological vectorspace V , there exists a projective diffeological vector space P ( V ) together with a linear subduction P ( V ) → V .Proof. Let V be an arbitrary diffeological vector space. We construct P ( V ) as the coproduct in D Vect of all F ( U ) indexed by all plots U → V . By the universal property of free diffeological vectorspaces, there is a canonical smooth linear map F ( U ) → V . Therefore, there is a smooth linear map P ( V ) → V . By construction, it is easy to see that this map is a subduction. By Corollary 6.4 andRemark 6.9(2), we know that P ( V ) is a projective diffeological vector space. (cid:3) Note that since the free functor F : D iff → D Vect is a left adjoint, P ( V ) constructed in the proofof the above theorem is actually a free diffeological vector space.Of course, given a diffeological vector space V , the projective diffeological vector space P ( V )constructed in the proof of the above theorem is functorial but huge. However, there is a naturaltransformation P →
1, and f : V → W is a linear subduction if and only if P ( f ) : P ( V ) → P ( W )is. Here is a non-functorial but much smaller construction, whose proof is similar to the proof of theabove theorem: Proposition 6.12.
Let V be a diffeological vector space, and let { p i : U i → V } be a generating setof the diffeology on V . Then P η i : ⊕ F ( U i ) → V is a linear subduction with ⊕ F ( U i ) a projectivediffeological vector space. Here is an immediate consequence:
Corollary 6.13.
Every projective diffeological vector space is a smooth direct summand of a coprod-uct of free diffeological vector spaces generated by open subsets of Euclidean spaces.
Proposition 6.14.
A diffeological vector space V is projective if and only if the functor L ∞ ( V, − ) : D Vect → D Vect preserves short exact sequences.Proof. ( ⇐ ) This is clear.( ⇒ ) Since D Vect has enough projectives, V is a retract in D Vect of P ( V ), which is introduced inthe proof of Theorem 6.11. Since the functor L ∞ ( W, − ) is a left adjoint for any diffeological vectorspace W and subductions are closed under retracts, we are left to show that L ∞ ( P ( V ) , − ) preservessubductions. By Remarks 3.6 and 6.5, we are left to show that for any U open subset of a Euclideanspace, the functor C ∞ ( U, − ) : D Vect → D Vect preserves linear subductions. This then follows fromCorollary 6.4. (cid:3)
As an easy corollary, we have:
Corollary 6.15. If V and W are projective diffeological vector spaces, then so is V ⊗ W . Homological algebra for diffeological vector spaces
In this section, we develop (relative) homological algebra for diffeological vector spaces basedon the results we get in the previous sections about linear subductions and projective diffeologicalvector spaces. We show that every diffeological vector space has a diffeological projective resolution,which is unique up to diffeological chain homotopy equivalence. Shanuel’s lemma, Horseshoe lemmaand (Short) Five lemma still hold in this setting. From a short exact sequence of diffeological(co)chain complexes, we get a long sequence in D Vect which is also exact in V ect. Then we defineext diffeological vector space Ext n ( V, W ) for any diffeological vector spaces
V, W and any n ∈ N . Forany short exact sequence of diffeological vector spaces in the first or second variable, we get a longsequence of ext diffeological vector spaces which is exact in V ect. Finally we show that Ext ( V, W )classifies all short exact sequences in D Vect of the form 0 → W → A → V → D Ch of diffeological chain complexes to be the full subcategoryof the functor category D Vect Z consisting of objects in which the composition of consecutive arrowsare 0, where Z is viewed as a poset of integers with the opposite ordering. The morphisms in D Chare called diffeological chain maps . For any n ∈ Z , there exists a functor H n : D Ch → D Vectdefined by H n ( V ) = ker( d n ) / Im( d n +1 ), where V n − V nd n o o V n +1 d n +1 o o is a piece in the diffeological chain complex V , both ker( d n ) and Im( d n +1 ) are equipped withthe sub-diffeologies of V n , and H n ( V ) is equipped with the quotient diffeology. We call H n ( V ) the n th homology of V . Two diffeological chain maps f , g : V → W are called diffeologically OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 17 chain homotopic if there are smooth linear maps h n : V n → W n +1 for all n ∈ Z such that f n − g n = h n − ◦ d V n + d W n +1 ◦ h n for each n ∈ Z . This gives an equivalence relation on D Ch( V , W ),called the diffeological chain homotopy equivalence , which is compatible with compositionsin D Ch. Write h DCh for the quotient category. Then the homology functors H n factors throughthe projection D Ch → h DCh. A diffeological chain map f : V → W is called a homologyisomorphism if for each n ∈ Z , H n ( f ) is an isomorphism in D Vect. A diffeological chain complex V is called exact at n th spot if the induced map 0 → V n − / ker( d n − ) → V n → Im( d n ) → V n − / ker( d n − ) equipped with the quotient diffeology of V n − and Im( d n ) equipped withthe sub-diffeology of V n +1 is a short exact sequence in D Vect. A diffeological chain complex iscalled exact if it is exact at every spot. A diffeological projective resolution of a diffeologicalvector space V is an exact diffeological chain complex V such that V − = V , V n = 0 for every n < −
1, and V n is projective for every n ≥
0. In this case, we write P ( V ) for the diffeologicalchain complex with P ( V ) n = V n for each n ≥ P ( V ) n = 0 for each n <
0. Then thediffeological chain map from P ( V ) to the diffeological chain complex with V concentrated at the 0 th position is a homology isomorphism. By abuse of notation, we also call this diffeological chain mapa diffeological projective resolution , and denote it by P ( V ) → V .Similarly, one can define the category D Ch o of diffeological cochain complexes and cohomologyfunctors H n : D Ch o → D Vect.As the proof of [We, Lemma 2.2.5], one can show that there is a diffeological projective resolutionfor every diffeological vector space. As the proof of [We, Comparison Theorem 2.2.6], for anydiffeological projective resolution V of V (or write as P ( V ) → V ), any exact diffeological chaincomplex W with W i = 0 for all i ≤ −
2, and any smooth linear map f : V → W − , there existsa diffeological chain map V → W extending f , and the corresponding diffeological chain map P ( V ) → ¯ W is unique up to diffeological chain homotopy equivalence, where ¯ W is derived from W by replacing W − by 0. In particular, for any two diffeological projective resolutions V and V ′ of V , there is an isomorphism between P ( V ) and P ′ ( V ) in h DCh from the corresponding diffeologicalchain maps between V and V ′ extending 1 V : V → V . Lemma 7.1 (Schanuel) . Given short exact sequences of diffeological vector spaces / / K i / / P π / / M / / and / / K ′ i ′ / / P ′ π ′ / / M / / with both P and P ′ projective, there is an isomorphism K ⊕ P ′ ∼ = K ′ ⊕ P in D Vect .Proof.
By assumption, we have the following commutative diagrams in D Vect0 / / K i / / α (cid:15) (cid:15) P π / / β (cid:15) (cid:15) M / / M (cid:15) (cid:15) / / K ′ i ′ / / ρ (cid:15) (cid:15) P ′ π ′ / / γ (cid:15) (cid:15) M / / M (cid:15) (cid:15) / / K i / / P π / / M / / . It is shown in the proof of [R, Proposition 3.12] that0 / / K ( i,α ) / / P ⊕ K ′ β − i ′ / / P ′ / / V ect, so we are left to show that this is actually a short exact sequencein D Vect. ( i, α ) is an induction since i is. To show that ( β − i ′ ) is a subduction, note that thereexists a smooth linear map δ ′ : P ′ → K ′ such that β ◦ γ − P ′ = i ′ δ ′ . For any plot p : U → P ′ , themap ( γ ◦ p, δ ′ ◦ p ) : U → P ⊕ K ′ is smooth, and we have p = ( β − i ′ ) ◦ ( γ ◦ p, δ ′ ◦ p ). (cid:3) Lemma 7.2 (Horseshoe) . Let (cid:15) (cid:15) · · · / / P ′ d ′ / / P ′ ǫ ′ / / A ′ i A (cid:15) (cid:15) / / A π A (cid:15) (cid:15) · · · / / P ′′ d ′′ / / P ′′ ǫ ′′ / / A ′′ / / (cid:15) (cid:15) be a diagram in D Vect with the two horizontal lines diffeological projective resolutions of A ′ and A ′′ respectively, and the vertical line a short exact sequence of diffeological vector spaces. Then thereexists a diffeological projective resolution of A in the middle horizontal line making the followingdiagram commutative and all vertical lines short exact sequences in D Vect0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) · · · / / P ′ i (cid:15) (cid:15) d ′ / / P ′ i (cid:15) (cid:15) ǫ ′ / / A ′ i A (cid:15) (cid:15) / / · · · / / P ′ ⊕ P ′′ π (cid:15) (cid:15) d / / P ′ ⊕ P ′′ π (cid:15) (cid:15) ǫ / / A π A (cid:15) (cid:15) / / · · · / / P ′′ d ′′ / / (cid:15) (cid:15) P ′′ (cid:15) (cid:15) ǫ ′′ / / A ′′ / / (cid:15) (cid:15)
00 0 0
Proof.
Since π A is a linear subduction and P ′′ is projective, there exists a smooth linear map f : P ′′ → A such that π A ◦ f = ǫ ′′ . Define ǫ := i A ◦ ǫ ′ + f . The inductive proof of [We, Horseshoe OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 19
Lemma 2.2.8] showed that 0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / ker( ǫ ′ ) / / (cid:15) (cid:15) P ′ ǫ ′ / / (cid:15) (cid:15) A ′ / / i A (cid:15) (cid:15) / / ker( ǫ ) / / (cid:15) (cid:15) P ′ ⊕ P ′′ ǫ / / (cid:15) (cid:15) A / / π A (cid:15) (cid:15) / / ker( ǫ ′′ ) / / (cid:15) (cid:15) P ′′ ǫ ′′ / / (cid:15) (cid:15) A ′′ / / (cid:15) (cid:15)
00 0 0is a commutative diagram with all vertical and horizontal lines short exact sequences in V ect, so weare left to show that ǫ : P := P ′ ⊕ P ′′ → A is a subduction and 0 → ker( ǫ ′ ) → ker( ǫ ) → ker( ǫ ′′ ) → p : U → P ′′ and q : U → A such that ǫ ′′ ◦ p = π A ◦ q , locally there exists a plot r : U → P such that ǫ ◦ r = q and π ◦ r = p . Since finite products and coproducts coincide in D Vect, thisfact follows from the assumptions that P ′ is a diffeological vector space, ǫ ′ is a subduction, and0 → A ′ → A → A ′′ → D Vect. (cid:3)
Theorem 7.3.
Let f : A → B and g : B → C be diffeological chain maps between diffeological chaincomplexes, such that for each n , / / A n f n / / B n g n / / C n / / is a short exact sequence in D Vect . Then we have the following sequence in D Vect which is exactin V ect : · · · / / H n ( A ) f ∗ / / H n ( B ) g ∗ / / H n ( C ) = Since every short exact sequence in D Vect is also a short exact sequence in V ect, we havethis long exact sequence in V ect. By functoriality of homology functors, we know that all f ∗ ’s and g ∗ ’s are smooth. So we are left to show that all connecting linear maps δ are smooth. Recall that δ n : H n ( C ) → H n − ( A ) is defined as follows:0 / / A n f n / / d An (cid:15) (cid:15) B n g n / / d Bn (cid:15) (cid:15) C n / / d Cn (cid:15) (cid:15) / / A n − f n − / / B n − g n − / / C n − / / c for [ c ] ∈ ker( d Cn ) / Im( d Cn +1 ) = H n ( C ). Since g n is surjective, we can pick b ∈ B n such that g n ( b ) = c . Then there exists a ∈ ker( d An − ) ⊆ A n − such that f n − ( a ) = d Bn ( b ).The map δ n is defined by δ n ([ c ]) = [ a ], which is independent of the choice of representative c of [ c ]and lift b ∈ B n . Any plot p : U → H n ( C ) globally lifts to a plot q : U → ker( d Cn ), since the quotientmap ker( d Cn ) → H n ( C ) is a linear subduction. If we write i : ker( d Cn ) → C n for the inclusion map,then the plot i ◦ q : U → C n globally lifts to a plot r : U → B n , since g n is a linear subduction. So by diagram chasing, the plot d Bn ◦ r : U → B n − globally lifts to a plot s : U → ker( d An − ). Thisproves the smoothness of δ n . (cid:3) Definition 7.4. Given diffeological vector spaces V and W , we define Ext n ( V, W ) := H n ( L ∞ ( P ( V ) , W )) , where P ( V ) → V is a diffeological projective resolution. Here are some basic properties for Ext n ( V, W ): Remark . (1) Ext n ( V, W ) does not depend on the choice of diffeological projective resolution of V .(2) Ext ( V, W ) is always isomorphic to L ∞ ( V, W ) in D Vect.(3) If V is projective, then Ext n ( V, W ) = 0 for all n ≥ injective diffeological vector space as dual of projective diffeological vectorspace, i.e., a diffeological vector space V is injective if and only if for every linear induction f : W → W and every smooth linear map g : W → V , there exists a smooth linearmap h : W → V such that g = h ◦ f . It is direct to show that if V is injective, thenExt n ( W, V ) = 0 for all n ≥ 1. In particular, Ext n ( W, R ind ) = 0 for all n ≥ { V i } i ∈ I ∪{ W } be a set of diffeological vector spaces. Then we have natural isomorphismsbetween Ext n ( ⊕ i ∈ I V i , W ) and Q i ∈ I Ext n ( V i , W ) in D Vect for all n ∈ N .To prove this, besides the usual diagram chasing in homological algebra, we also need thefollowing facts in D Vect (and D iff): • Any (co)product of linear subductions is again a linear subduction. For the case ofcoproduct, it can be proved by the description of the coproduct diffeology from theproof of Proposition 3.2. • Let X j be a subset of a diffeological space Y j for each j ∈ J . Then the sub-diffeologyon Q j ∈ J X j from Q j ∈ J Y j coincides with the product diffeology with each X j equippedwith the sub-diffeology from Y j . • Let A i be a linear subspace of V i for each i ∈ I . Then we have a natural isomorphismbetween ( Q i ∈ I V i ) / ( Q i ∈ I A i ) and Q i ∈ I ( V i /A i ) in D Vect.(6) Let { V }∪{ W j } j ∈ J be a set of diffeological vector spaces. Then we have natural isomorphismsbetween Ext n ( V, Q j ∈ J W j ) and Q j ∈ J Ext n ( V, W j ) in D Vect for all n ∈ N .(7) (Dimension shift) Let V P d o o P d o o · · · o o P n − d n − o o W d n o o be a finite diffeological chain complex such that it is exact at every middle spot, each P i is projective, d is a linear subduction and d n is a linear induction. Then Ext m + n ( V, A ) isisomorphic to Ext m ( W, A ) in D Vect for any m ∈ Z + and any diffeological vector space A . Theorem 7.6. Let / / W i / / W π / / W / / be a short exact sequence in D Vect .Then for any diffeological vector space V , we have the following sequence in D Vect which is exactin V ect : / / L ∞ ( V, W ) i ∗ / / L ∞ ( V, W ) π ∗ / / L ∞ ( V, W ) = Proof. Take a diffeological projective resolution P ( V ) → V . Since0 / / W i / / W π / / W / / D Vect and each P j is projective, by Proposition 6.14, we get diffeologicalcochain maps i ∗ : L ∞ ( P ( V ) , W ) → L ∞ ( P ( V ) , W )and π ∗ : L ∞ ( P ( V ) , W ) → L ∞ ( P ( V ) , W )between diffeological cochain complexes such that for each n / / L ∞ ( P n , W ) i ∗ / / L ∞ ( P n , W ) π ∗ / / L ∞ ( P n , W ) / / D Vect. The result then follows from the cohomological version ofTheorem 7.3. (cid:3) Dually, we have Theorem 7.7. Let / / V i / / V π / / V / / be a short exact sequence in D Vect . Thenfor any diffeological vector space W , we have the following sequence in D Vect which is exact in V ect : / / L ∞ ( V , W ) π ∗ / / L ∞ ( V , W ) i ∗ / / L ∞ ( V , W ) = This follows directly from the Horseshoe lemma and Theorem 7.3. (cid:3) Corollary 7.8. Let V and W be diffeological vector spaces. If Ext ( W, V ) = 0 , then every shortexact sequence → V → A → W → in D Vect splits smoothly.Proof. By Theorem 7.6, we know that L ∞ ( W, A ) → L ∞ ( W, W ) is surjective. The result then followsfrom Theorem 3.16. (cid:3) Hence, with the notations in Example 4.3, Ext ( Q ω R , K ) = 0. Corollary 7.9. A diffeological vector space V is projective if and only if Ext ( V, W ) = 0 for every diffeological vector space W .Proof. ( ⇒ ) This follows from Remark 7.5(3).( ⇐ ) This follows from Corollary 7.8 and Remark 6.9(1). (cid:3) As an easy corollary, we know that if V → W is a linear subduction between projective diffeologicalvector spaces, then its kernel is also projective. In particular, if f : M → N is a smooth fiber bundlebetween smooth manifolds, then the kernel of F ( f ) : F ( M ) → F ( N ) is a projective diffeologicalvector space.Finally we are going to prove the inverse of Corollary 7.8. For V, W diffeological vector spaces, we define two short exact sequences 0 → V → A → W → → V → A ′ → W → D Vect to be equivalent if there is a commutative diagram in D Vect0 / / V / / V (cid:15) (cid:15) A / / f (cid:15) (cid:15) W / / W (cid:15) (cid:15) / / V / / A ′ / / W / / f an isomorphism in D Vect. (We will see in the next lemma that indeed only the existence of f is needed.) This defines an equivalence relation on the set of all short exact sequences in D Vectof the form 0 → V → A → W → 0, and we write the quotient set as e ( W, V ). Lemma 7.10. Given a commutative diagram in D Vect0 / / V i / / g (cid:15) (cid:15) A π / / f (cid:15) (cid:15) W / / h (cid:15) (cid:15) / / V ′ i ′ / / A ′ π ′ / / W ′ / / with both rows short exact sequences in D Vect , and both g and h isomorphisms in D Vect . Then f is also an isomorphism in D Vect .Proof. From [H, The Short Five Lemma 1.17], we know that f is an isomorphism in V ect. We areleft to show that f − is smooth. For any plot p : U → A ′ , since h − ◦ π ′ ◦ p : U → W is smooth and π is a linear subduction, we have a smooth linear map q : U → A such that π ◦ q = h − ◦ π ′ ◦ p . Then π ◦ ( f − ◦ p − q ) = 0 implies that π ′ ◦ ( p − f ◦ q ) = 0. Since the bottom row in the above diagram isshort exact in D Vect, there exists a smooth linear map r : U → V ′ such that f ◦ q − p = i ′ ◦ r . So f − ◦ p = q + f − ◦ i ′ ◦ r = q + i ◦ g − ◦ r is smooth, which implies that f − is smooth. (cid:3) Slightly more generally, we can prove the following in a similar way: Proposition 7.11. Let V / / f (cid:15) (cid:15) V / / f (cid:15) (cid:15) V d / / f (cid:15) (cid:15) V / / f (cid:15) (cid:15) V f (cid:15) (cid:15) W / / W ∂ / / W / / W / / W be a commutative diagram in D Vect , such that both rows are exact in V ect , f is surjective, f isinjective, and both f and f are isomorphisms in D Vect . If the induced maps V → Im( d ) is asubduction and W / ker( ∂ ) → W is an induction, then f is an isomorphism in D Vect . Theorem 7.12. There is a bijection between e ( W, V ) and Ext ( W, V ) which sends the class ofsmoothly split short exact sequences in e ( W, V ) to in Ext ( W, V ) .Proof. As explained in [R, pp. 422-423], the map ψ : e ( W, V ) → Ext ( W, V ) is defined as follows:For any short exact sequence 0 → V → A → W → D Vect and any diffeological projectiveresolution P ( W ) → W , we have a commutative diagram (not necessarily unique) in D Vect · · · / / P / / (cid:15) (cid:15) P / / f (cid:15) (cid:15) P / / (cid:15) (cid:15) W / / W (cid:15) (cid:15) / / V / / A / / W / / . Then ψ takes the class of 0 → V → A → W → e ( W, V ) to [ f ] ∈ Ext ( W, V ). As shown in [R]that this map depends neither on representing short exact sequence in e ( W, V ) nor on diffeological OMOLOGICAL ALGEBRA FOR DIFFEOLOGICAL VECTOR SPACES 23 projective resolution of W nor on the choice of f . The second statement of the theorem can thenbe proved as [R, Lemma 7.27].The inverse map θ : Ext ( W, V ) → e ( W, V ) can be constructed as follows: For any diffeologicalprojective resolution P ( W ) → W and any f ∈ L ∞ ( P , V ) with d ∗ ( f ) = 0, let A be the pushout of V P d / / f o o P in D Vect. As proved in [R, Lemma 7.28] that we have a short exact sequence0 / / V i / / A π / / W / / V ect. It is easy to check that π is a subduction since P → W is, and i is an induction since A is a pushout, ker( d ) → P is an induction, and P → ker( d ) is a subduction. As shown in [R,Theorem 7.30] that θ is independent of the choice of representative of [ f ] ∈ Ext ( W, V ), and θ is theinverse of ψ . (cid:3) As an immediate corollary to Theorem 7.12 and Corollary 7.8, we know that for any fixeddiffeological vector spaces V and W , Ext ( W, V ) = 0 if and only if every short exact sequence0 → V → A → W → D Vect splits smoothly. 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