Pushforward and projective diffeological vector pseudo-bundles
aa r X i v : . [ m a t h . DG ] F e b PUSHFORWARD AND PROJECTIVE DIFFEOLOGICAL VECTORPSEUDO-BUNDLES
ENXIN WU
Abstract.
In this paper, we study a new operation named pushforward on diffeologicalvector pseudo-bundles, which is left adjoint to the pullback. We show how to pushfor-ward projective diffeological vector pseudo-bundles to get projective diffeological vectorspaces, producing many concrete new examples. This brings new objects to diffeologyfrom classical vector bundle theory. Introduction
Diffeological spaces are elegant generalizations of smooth manifolds, including infinite-dimensional spaces like mapping spaces and diffeomorphism groups, and singular spaceslike smooth manifolds with boundary or corners, orbifolds and irrational tori.On diffeological spaces, one can still do some differential geometry and topology, suchas differential forms and tangent bundles. These tangent bundles are in general no longerlocally trivial. Instead, they are diffeological vector pseudo-bundles. We studied theseobjects and operations on them in [CWp], on which the current paper is based.On the other hand, the theory of diffeological vector spaces and their homological algebrais intimately related to analysis and geometry; see [W, CW16, CWa]. The projective objectsthere deserve special attention. However, in general neither is it easy to test whether a givendiffeological vector space is projective or not, nor is it straightforward to construct manyconcrete projective objects.In this paper, we propose a way to use diffeological vector pseudo-bundles to study dif-feological vector spaces. We generalize some results of projective objects for diffeologicalvector spaces to such bundles. In particular, we show that every classical vector bundle issuch a projective object. We introduce a left adjoint called pushforward to the pullbackon diffeological vector pseudo-bundles, and we show that the free diffeological vector spacegenerated by a diffeological space has a bundle-theorecical explanation, and that pushfor-ward preserves projectives. In this way, we construct many concrete projective diffeologicalvector spaces from classical vector bundle theory.Here is the structure of the paper. In Section 2, we briefly review some necessary back-ground. In Section 3, we introduce pushforward on diffeological vector pseudo-bundles.Section 4 contains three parts, including necessary and sufficient conditions of splitting ofshort exact sequences of diffeological vector pseudo-bundles, examples and properties of theprojective objects, and preservation of projectives by pushforward. In particular, we getmany new examples of projective diffeological vector spaces from classical vector bundles.
Date : February 16, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Diffeological vector pseudo-bundle, diffeological vector space, pushforward. Background
We give a very brief review together with many related references in this section.
Definition 2.1. A diffeological space is a set X together with a collection of maps U → X (called plots ) from open subsets U of Euclidean spaces, such that (1) every constant map is a plot; (2) V → U → X is a plot if the first one is a smooth map between open subsets ofEuclidean spaces and the second one is a plot; (3) U → X is a plot if and only if there is an open cover of U such that each restrictionis a plot.A smooth map X → Y between diffeological spaces is a map which sends plots of X toplots of Y . Diffeological spaces with smooth maps form a category denoted Diff . The idea of a diffeological space was introduced in [S]. [I13] is currently the standardreference for the subject. Also see [CSW, Section 2] for a concise summary for the basicsof diffeological spaces.The category Diff has excellent properties. It contains the category of smooth manifoldsas a full subcategory, and it is complete, cocomplete and cartesian closed. In particular,we have subspaces, quotient spaces and mapping spaces for diffeological spaces. Like chartsfor manifolds, we have various generating sets of plots for a diffeological space. Everydiffeological space has a canonical topology called the D -topology; see [I85, CSW]. Everydiffeological space has a tangent bundle; see [H, CW16, CW17]. Diffeological vector spacesare the vector space objects in Diff. Every vector space can be equipped with a smallestdiffeology called the fine diffeology, making it a diffeological vector space; see [I07]. Thereare many other kinds of diffeological vector spaces in practice. Hierachies of diffeologicalvector spaces were studied in [CW19], and homological algebra of diffeological vector spaces,including free and projective objects, were introduced in [W].We recall the following concepts from [CWp]: Definition 2.2. A diffeological vector pseudo-bundle over a diffeological space B is asmooth map π : E → B between diffeological spaces such that the following conditions hold: (1) for each b ∈ B , π − ( b ) =: E b is a vector space; (2) the fibrewise addition E × B E → E and the fibrewise scalar multiplication R × E → E are smooth; (3) the zero section σ : B → E is smooth. Definition 2.3.
Given a diffeological space B , a bundle map over B is a commutativetriangle E f / / π (cid:25) (cid:25) ✷✷✷✷✷✷ E π (cid:5) (cid:5) ☞☞☞☞☞☞ B, where π , π are diffeological vector pseudo-bundles over B , f is smooth and for each b ∈ B ,the restriction f | E ,b : E ,b → E ,f ( b ) is linear.Such f is called a bundle subduction (resp. bundle induction ) over B if it is both abundle map over B and a subduction (resp. an induction), i.e., it is equivalent to a quotientmap (resp. an inclusion of a subspace). USHFORWARD AND PROJECTIVE DIFFEOLOGICAL VECTOR PSEUDO-BUNDLES 3
For a fixed diffeological space B , all diffeological vector pseudo-bundles over B and bundlemaps over B form a category, denoted DVPB B . An isomorphism in DVPB B is called a bundle isomorphism over B . A bundle map over B is a bundle isomorphism if and onlyif it is both a bundle induction and a bundle subduction over B . Definition 2.4.
A commutative square E g / / π (cid:15) (cid:15) E ′ π ′ (cid:15) (cid:15) B f / / B ′ in Diff with π and π ′ being diffeological vector pseudo-bundles, is called a bundle map , iffor each b ∈ B , g | E b : E b → E ′ f ( b ) is linear. All diffeological vector pseudo-bundles and bundle maps form a category denoted byDVPB.Note that diffeological vector pseudo-bundles are neither diffeological fibre bundles in [I85,I13], nor diffeological fibrations in [CW14]. They were introduced to encode tangent bundlesof diffeological spaces ([CW16]). Many operations on DVPB B and DVPB were studiedin [CWp]. 3. Pushforward
Recall from [CWp, Section 3.1] that one can pullback diffeological vector pseudo-bundlesvia smooth maps, i.e., a smooth map f : B → B ′ induces a functor f ∗ : DVPB B ′ → DVPB B by pullback. Now we define a related operation as follows:Given a smooth map f : B → B ′ and a diffeological vector pseudo-bundle π : E → B ,we define E ′ = (cid:0) a b ′ ∈ Im( f ) ( M b ∈ f − ( b ′ ) E b ) (cid:1) a ( a b ′ / ∈ Im( f ) R ) . (1)There are canonical maps π f : E ′ → B ′ sending the fibre above b ′ to b ′ , and α f : E → E ′ with E b ֒ → L ˜ b ∈ f − ( f ( b )) E ˜ b . We then have a natural commutative square E α f / / π (cid:15) (cid:15) E ′ π f (cid:15) (cid:15) B f / / B ′ . Hence, we can equip E ′ with the dvsification of the diffeology generated by the upperhorizontal map α f of the above square via [CWp, Proposition 3.3], making the right verticalmap π f a diffeological vector pseudo-bundle over B ′ . (As a warning , each fibre of E ′ maynot be the direct sum of those of E as diffeological vector spaces; see Proposition 3.5.) Moreprecisely, we have the following explicit description of a generating set of plots on E ′ : Lemma 3.1.
A plot on E ′ is locally of one of the following forms: (1) U → E ′ defined by a finite sum P i α f ◦ p i , where p i : U → E are plots on E suchthat all f ◦ π ◦ p i ’s match; (2) the composite of a plot of B ′ followed by the zero section B ′ → E ′ . ENXIN WU
Proof.
This is straightforward from the description of dvsification in [CWp]. (cid:3)
It is straightforward to check that we get a functor f ∗ : DVPB B → DVPB B ′ , called the pushforward of f , and we write E ′ above as f ∗ ( E ). Moreover, from the above lemma, wehave(1) f ′∗ ◦ f ∗ = ( f ′ ◦ f ) ∗ for any smooth maps f : B → B ′ and f ′ : B ′ → B ′′ ;(2) (1 B ) ∗ = the identity on DVPB B .Here is the key result for pushforward: Theorem 3.2.
Given a smooth map f : B → B ′ , we have an adjoint pair of functors f ∗ : DVPB B ⇋ DVPB B ′ : f ∗ . Proof.
We show that there is a natural bijection DVPB B ( E, f ∗ ( E ′ )) ∼ = DVPB B ′ ( f ∗ ( E ) , E ′ ).Given a bundle map E → f ∗ ( E ′ ) over B , we have E b → E ′ f ( b ) for each b ∈ B , which induce L b ∈ f − ( b ′ ) E b → E ′ b ′ , and hence a map f ∗ ( E ) → E ′ . This is clearly a bundle map over B ′ .Conversely, given a bundle map f ∗ ( E ) → E ′ over B ′ , we have a map L b ∈ f − ( b ′ ) E b → E ′ b ′ foreach b ′ ∈ Im( f ). It then induces a map E b → E ′ f ( b ) , which together give a map E → f ∗ ( E ′ ).It is straightforward to check that this is a bundle map over B . These procedures are inversesto each other, and therefore we proved the desired result. (cid:3) Remark . One can change the direct sum in the construction (see (1)) of the pushforwardto the direct product. However, it does not seem possible to give a suitable diffeology onthe resulting space such that it is a functor in a natural way which is right adjoint to thepullback.We have the following bundle-theoretical explanation of a free diffeological vector spaceintroduced in [W]:
Proposition 3.4.
For any diffeological space B , the total space of the pushforward of thetrivial bundle B × R → B along the map B → R is the free diffeological vector space F ( B ) .Proof. This follows directly from the diffeology of the total space of the pushforward (seeLemma 3.1) and the diffeology on free diffeological vector space (see proof of [W, Proposi-tion 3.5]). (cid:3)
From [CWp, Section 3], we know that the usual operations on diffeological vector pseudo-bundles have the desired diffeology on each fibre. But pushforward is an exception:
Proposition 3.5.
Let f : B → B ′ be a smooth map, and let E → B be a diffeological vectorpseudo-bundle. Then the diffeology on the fibre at b ′ of the pushforward f ∗ ( E ) has the directsum diffeology of the diffeological vector spaces E b ’s with f ( b ) = b ′ if and only if f − ( b ′ ) asa subspace of B has the discrete diffeology.Proof. This follows directly from Lemma 3.1. (cid:3)
Here is the universal property for pushforward:
Proposition 3.6.
Given a bundle map E f / / π (cid:15) (cid:15) E ′ π ′ (cid:15) (cid:15) B g / / B ′ , USHFORWARD AND PROJECTIVE DIFFEOLOGICAL VECTOR PSEUDO-BUNDLES 5 there exists a unique bundle map β : g ∗ ( E ) → E ′ over B ′ such that f = β ◦ α g .Proof. This is clear by the construction of pushforward. (cid:3)
Pushforward could send non-isomorphic bundles to isomorphic ones:
Example . Write B for the cross with the gluing diffeology, and write B ′ for the crosswith the subset diffeology of R . Then B → B ′ defined as the identity underlying setmap is smooth, but its inverse is not; see [CW16, Example 3.19]. We show below that theinduced map F ( B ) → F ( B ′ ), which is identity for the underlying vector spaces, is indeedan isomorphism of diffeological vector spaces. This means that the pushforward of the twotrivial bundles B × R → B and B ′ × R → B ′ along the maps B → R and B ′ → R areisomorphic, but clearly the two bundles are not.By definition of a free diffeological vector space, every plot p : U → F ( B ′ ) can belocally written as a finite sum p ( u ) = P i r i ( u )( p i ( u ) , p i ( u )) for smooth maps r i , p i , p i with codomain R satisfying p i ( u ) p i ( u ) = 0 for all u . It is enough to show that p canbe viewed as a plot of F ( B ). This is the case since ( p i ( u ) , p i ( u )) can be written as( p i ( u ) ,
0) + (0 , p i ( u )) − (0 ,
0) each term viewed as a plot of B .4. Projective diffeological vector pseudo-bundles
Enough projectives.
In this subsection, we will work in the category DVPB B for afixed diffeological space B . So we will omit the phrase ‘over B ’ in many places as long asno confusion shall occur. Note that when we take B = R , we recover the correspondingresults for the category of diffeological vector spaces.We first study smooth splitting of diffeological vector pseudo-bundles, which will be usedlater in the paper. Definition 4.1.
A diagram of morphisms E f / / E g / / E in DVPB B , is called a short exact sequence if f is a bundle induction, g is a bundlesubduction, and E ,b f b / / E ,b g b / / E ,b is exact (i.e., ker( g b ) = Im( f b ) ) for every b ∈ B . As a direct consequence of the above definition, we have:
Corollary 4.2.
Given a short exact sequence E / / E / / E of diffeological vector pseudo-bundles over B , we have a bundle isomorphism E /E ∼ = E over B . The splitting of a short exact sequence goes as usual:
Theorem 4.3.
Assume that E f / / E g / / E is a short exact sequence of diffeological vector pseudo-bundles over B . Then the followingare equivalent: (1) there exists a bundle map g ′ : E → E over B such that g ◦ g ′ = 1 E ; ENXIN WU (2) there exists a bundle map f ′ : E → E over B such that f ′ ◦ f = 1 E ; (3) there exists a bundle isomorphism E → E ⊕ E over B making the followingdiagram commutative: E f / / = (cid:15) (cid:15) E g / / (cid:15) (cid:15) E (cid:15) (cid:15) E i / / E ⊕ E p / / E If any one of the conditions holds in the theorem, we say that the short exact sequence splits smoothly , and that E (resp. E ) is a smooth direct summand of E . Althoughevery short exact sequence of vector spaces splits, it is not the case in DVPB B , even when B = R ; see [W, Example 4.3] or [CW19, Example 4.1]. Proof.
We show below that (1) ⇔ (3), and (2) ⇔ (3) can be proved similarly.(1) ⇒ (3): since we have bundle maps f : E → E and g ′ : E → E , we define E ⊕ E → E by ( x , x ) f ( x ) + g ′ ( x ) for any x ∈ E ,b , x ∈ E ,b and b ∈ B . Thisis clearly a bundle map over B . Its inverse is given by x ( f − ( x − g ′ ◦ g ( x )) , g ( x )). It isstraightforward to check that this is well-defined, and it is smooth since f is an induction.(3) ⇒ (1): g ′ is defined by the composite E i / / E ⊕ E ∼ = / / E . The rest arestraightforward to check. (cid:3) Now we can define projective diffeological vector pseudo-bundles, and show that thereare enough such objects.
Definition 4.4.
A diffeological vector pseudo-bundle E → B is called projective if for anybundle subduction f : E → E over B and any bundle map g : E → E over B , there existsa bundle map h : E → E over B making the triangle commutate: E g (cid:15) (cid:15) h } } E f / / E . Formally, we have the following basic properties:
Proposition 4.5. (1)
Each diffeological vector pseudo-bundle E i → B is projective if and only if the directsum L i E i → B is projective. (2) Projectives are closed under taking retracts. (3)
Every bundle subduction to a projective splits smoothly.
Recall from [CWp, Section 3.2.5] that given a smooth map X → B , we get a diffeologicalvector pseudo-bundle F B ( X ) → B . Lemma 4.6.
Let X → B be a smooth map. The corresponding diffeological vector pseudo-bundle F B ( X ) → B is projective if and only if for every bundle subduction f : E → E over B and any smooth map g : X → E over B , there exists a smooth map h : X → E over B such that g = f ◦ h .Proof. As usual, this follows from the universal property of F B ( X ) → B ; see [CWp, Sec-tion 3.2.5]. (cid:3) USHFORWARD AND PROJECTIVE DIFFEOLOGICAL VECTOR PSEUDO-BUNDLES 7
Proposition 4.7.
Every plot U → B induces a projective diffeological vector pseudo-bundle F B ( U ) → B .Proof. Given any bundle subduction f : E → E over B and any smooth map g : U → E over B , we have smooth local liftings h i of g to E . Let { λ i } be a smooth partition of unitysubordinate to the corresponding open cover { U i } of U . Then P i λ i · h i : U → E is aglobal smooth lifting of g over B , where each λ i · h i : U → E is defined as( λ i · h i )( u ) = ( λ i ( u ) h i ( u ) , if u ∈ U i σ ◦ π ◦ g ( u ) , elsewith σ : B → E the zero section and π : E → B the given diffeological vector pseudo-bundle. The result then follows from Lemma 4.6. (cid:3) As a direct consequence of the above proof, we have:
Corollary 4.8.
For every bundle subduction, a plot of the total space of the codomain globally lifts to a plot of the total space of the domain.
Theorem 4.9.
For every diffeological space B , the category DVPB B has enough projec-tives, i.e., given any diffeological vector pseudo-bundle E → B , there exists a projectivediffeological vector pseudo-bundle E ′ → B together with a bundle subduction E ′ → E over B .Proof. We take E ′ → B to be the direct sum in DVPB B of all F B ( U ) → B ’s indexedover all plots U → E . By Proposition 4.7, each F B ( U ) → B is projective, and hence byProposition 4.5(1), E ′ is projective. By the universal property of F B ( U ) → B , we get abundle map F B ( U ) → E over B , and hence a bundle map E ′ → E over B . By construction,this map is a subduction. (cid:3) Examples and properties of projectives.
We first give some examples of projectivediffeological vector pseudo-bundles related to classical vector bundle theory. To do so, weneed:
Lemma 4.10.
For a smooth map f : B → B ′ , the pullback f ∗ sends a bundle subductionover B ′ to a bundle subduction over B , and hence it preserves short exact sequences.Proof. Let g : E ′ → E ′ be a bundle subduction over B ′ . Then f ∗ ( E ′ ) → f ∗ ( E ′ ) is given bysending ( b, x ) to ( b, g ( x )). Every plot p : U → f ∗ ( E ′ ) gives rise to smooth maps p : U → B and p : U → E ′ via composition with the two projections. Since g is a bundle subduction, p locally lifts as a smooth map to E ′ , which together with p induces a local lifting of p to f ∗ ( E ′ ), showing the first claim.Since f ∗ is a right adjoint by Theorem 3.2, it preserves bundle inductions, which togetherwith the first claim proves the second one. (cid:3) Projectiveness is local in the following sense:
Proposition 4.11.
Let π : E → B be a diffeological vector pseudo-bundle. Assume thatthere exists a D -open cover { B j } of B such that i ∗ j ( E ) → B j is projective in DVPB B j foreach j , where i j : B j → B denotes the inclusion, together with a smooth partition of unity { λ j : B → R } subordinate to this cover. Then π is projective in DVPB B . ENXIN WU
Proof.
For any bundle subduction f : E → E over B and any bundle map g : E → E over B , we get a diagram over B j for each j : i ∗ j ( E ) i ∗ j ( g ) (cid:15) (cid:15) i ∗ j ( E ) i ∗ j ( f ) / / i ∗ j ( E ) . Lemma 4.10 shows that the horizontal arrow is a bundle subduction over B j . By assumption,we have a smooth lifting h j : i ∗ j ( E ) → i ∗ j ( E ) over B j . Then P j λ j · h j : E → E is thebundle map over B as we desired. (cid:3) We also have the following expected result:
Proposition 4.12.
Let V be a projective diffeological vector space, and let B be a smoothmanifold. Then the trivial bundle B × V → B is projective. Surprisingly, note that the result can fail if B is an arbitrary diffeological space; seeExample 4.25. Proof.
We first reduce the above statement to a special case. By Proposition 4.11, it isenough to prove this for the case when B is an open subset of a Euclidean space. Recallthat every projective diffeological vector space is a retract of direct sums of F ( U )’s for opensubsets U of Euclidean spaces ([W, Corollary 6.15]). By Proposition 4.5(1) and (2), it isenough to show this for the case when V = F ( U ) for an open subset U of a Euclidean space.Now we prove the statement for the special case when V = F ( U ), and B, U are Euclideanopen subsets. As diffeological vector pseudo-bundles over B , we have isomorphisms F B ( B × U ) ∼ = B × F ( U ) of total spaces. The result then follows directly from Proposition 4.7. (cid:3) Combining the above two propositions together with the fact that every fine diffeologicalvector space is projective, we get:
Corollary 4.13.
Vector bundles in classical differential geometry are projective.
However, a projective diffeological vector pseudo-bundle does not need to be locallytrivial, even when the base space is Euclidean:
Example . Let f : R → R be the square function x x . By Proposition 4.7, F R ( R ) → R is projective. Clearly, the fibre is R for b < R for b = 0 and R for b >
0. Therefore,a projective diffeological vector pseudo-bundle does not need to be locally trivial.Now we discuss some properties of projective diffeological vector pseudo-bundles.
Proposition 4.15.
Every projective diffeological vector pseudo-bundle E → B is a retractof direct sum in DVPB B of F B ( U ) → B induced by some plots U → B .Proof. By the proof of Theorem 4.9, we get a bundle subduction E ′ → E over B with E ′ adirect sum in DVPB B of F B ( U ) → B induced by the plots U → E (and hence some plots U → B ). Since E → B is projective, the result then follows from Proposition 4.5(3). (cid:3) Using notations from [CW19], we have
Corollary 4.16.
Let E → B be a projective diffeological vector pseudo-bundle. Then E b ∈ SV for every b ∈ B , i.e., the smooth linear functionals on E b separate points. USHFORWARD AND PROJECTIVE DIFFEOLOGICAL VECTOR PSEUDO-BUNDLES 9
Proof.
By Proposition 4.15, we know that E is a retract of direct sum in DVPB B of F B ( U ) → B induced by some plots U → B . As SV is closed under taking retracts anddirect sums ([CW19, Proposition 3.11]), it is enough to show the claim for the special case F B ( U ) → B induced by a plot p : U → B . In this case, the fibre at b ∈ B is the freediffeological vector space generated by p − ( b ) ([CWp, Section 3.2.5]), which is a subset ofa Euclidean space, and hence p − ( b ) ∈ SD ′ , i.e., the smooth functions on p − ( b ) separatepoints. The result then follows from [CW19, Proposition 3.13]. (cid:3) One would expect that each fibre of a projective diffeological vector pseudo-bundle isa projective diffeological vector space. This is equivalent to the statement that the freediffeological vector space generated by any subset with the subset diffeology of a Euclideanspace is projective, by a similar argument as above. But I don’t know whether this is trueor not. Nevertheless, we have:
Proposition 4.17.
Let B be a diffeological space. Then every fibre of a projective diffeo-logical vector pseudo-bundle E → B is a projective diffeological vector space if and only iffor every plot p : U → B and every b ∈ B , the free diffeological vector space generated by p − ( b ) is projective.Proof. ( ⇒ ) This follows directly from Proposition 4.7.( ⇐ ) The proof follows from a similar argument as the one in the proof of the abovecorollary. (cid:3) Proposition 4.18.
Let B be a discrete diffeological space, i.e., every plot is locally constant.Then a diffeological vector pseudo-bundle over B is projective if and only if each fibre is aprojective diffeological vector space.Proof. ( ⇒ ) This follows from the definition of a discrete diffeological space, together withProposition 4.17 and [W, Corollary 6.4].( ⇐ ) This follows from the fact that every diffeological vector pseudo-bundle over adiscrete diffeological space is a coproduct in DVPB of diffeological vector spaces over apoint. (cid:3) Also, we have the following results:
Proposition 4.19.
Let π : E → B be a projective diffeological vector pseudo-bundle,and let π → π → π be a short exact sequence in DVPB B , with π i : E i → B . Then Hom B ( π, π ) → Hom( π, π ) → Hom( π, π ) is also a short exact sequence in DVPB B .Proof. By Proposition 4.15, we know that π is a retract of direct sum of F B ( U ) → B ’sindexed by some plots U → B . It is straightforward to check that retract and direct prod-uct preserve short exact sequences in DVPB B . For the direct product case, one needsCorollary 4.8 for the subduction part. By the universal property of free bundle inducedby a smooth map ([CWp, Section 3.2.5]), one has a bundle isomorphism over B fromHom B ( F B ( U ) , E i ) to the set Hom B ( U, E i ) of all smooth maps U → E i preserving B ,equipped with the subset diffeology of C ∞ ( U, E i ). Again by Corollary 4.8, it is directto check that Hom B ( U, ?) preserves short exact sequences in DVPB B . The result thenfollows by the above observations together with the first isomorphism in [CWp, Proposi-tion 3.13] (cid:3) Remark . The converse of Proposition 4.19 is false. This is because Hom B ( π, ?) alwayspreserves short exact sequences in DVPB B for the trivial bundle π : B × R → B , as it is naturally isomorphic to the identity functor. But the trivial bundle may not be projective;see Example 4.25.As a consequence of Proposition 4.19 and [CWp, Proposition 3.12], we have: Corollary 4.21. If E → B and E → B are projective diffeological vector pseudo-bundles,then so is their tensor product E ⊗ E → B . Base change.Theorem 4.22.
The pushforward f ∗ : DVPB B → DVPB B ′ sends projectives in the domainto the projectives in the codomain.Proof. By the adjunction of Theorem 3.2, the following lifting problems are equivalent: f ∗ ( E ) (cid:15) (cid:15) | | E ′ / / E ′ ⇐⇒ E (cid:15) (cid:15) y y f ∗ ( E ′ ) / / f ∗ ( E ′ ) , where E ′ → E ′ is a bundle subduction over B ′ . By Lemma 4.10 and Definition 4.4, weknow that the lifting problem on the right has a solution, and hence so is the one on theleft. (cid:3) This theorem has several applications. We first give another class of examples of projec-tive diffeological vector pseudo-bundles from tangent bundles of diffeological spaces. To doso, we need the following result:Note that projective diffeological vector pseudo-bundles are defined in DVPB B , but theyhave the similar property in DVPB as follows: Proposition 4.23.
Given a bundle subduction f : E ′ → E ′ over B ′ and a bundle map E g / / π (cid:15) (cid:15) E ′ (cid:15) (cid:15) B l / / B ′ with π projective, there exists a bundle map h : E → E ′ such that g = f ◦ h .Proof. By the universal property of pushforward (Proposition 3.6), we can write g as abundle map ˜ g : l ∗ ( E ) → E ′ over B ′ followed by the bundle map α l : E → l ∗ ( E ). ByTheorem 4.22, the assumption that π is projective over B implies that π l : l ∗ ( E ) → B ′ isprojective over B ′ . Therefore, we have a bundle map ˜ h : l ∗ ( E ) → E ′ over B ′ such that˜ g = f ◦ ˜ h . Then the composite ˜ h ◦ α l is the bundle map h we are looking for. (cid:3) Recall from [CW16, Theorem 4.17] that every tangent bundle T dvs B → B of a diffeolog-ical space B is a colimit in DVPB of the tangent bundles T U → U indexed by the plots U → B . Each T U → U is projective by Corollary 4.13. It is possible that some tangentbundles are projective. (But this is not always the case; see Example 4.25.) We show thisby an example: Example . Write B for the cross with the gluing diffeology. We show below that thetangent bundle T dvs B → B is projective. USHFORWARD AND PROJECTIVE DIFFEOLOGICAL VECTOR PSEUDO-BUNDLES 11
Note that B is the pushout of R R o o / / R in Diff. It is straightforward to check that the tangent bundle T dvs B → B is the colimit of T R (cid:15) (cid:15) T R T o o T / / (cid:15) (cid:15) T R (cid:15) (cid:15) R R o o / / R in DVPB. Write T x : T R → T dvs B and T y : T R → T dvs B for the two structural maps.Given a bundle subduction f : E → E over B and a bundle map g : T dvs B → E , since T R → R is projective, by Proposition 4.23 we have bundle maps hx, hy : T R → E suchthat g ◦ T x = f ◦ hx and g ◦ T y = f ◦ hy . By the universal property of pushout, we get adesired bundle map h : T dvs B → E over B with the required property.As another consequence of Theorem 4.22, we have the following example which givescounterexamples to several arguments: Example . If the free diffeological vector space F ( B ) is not projective, then the trivialbundle B × R → B is not projective. This happens when the D -topology on B is notHausdorff ([CW19, Corollary 3.17]). The proof of the statement follows from Proposition 3.4and Theorem 4.22.This example shows that not every trivial bundle is projective, even when the fibre is aprojective (or fine) diffeological vector space. It also shows that the pullback functor does not preserve projectives, since the trivial bundle B × R → B is the pullback of R → R along the map B → R . Furthermore, it shows that not every tangent bundle is projective.For example, T B → B is not projective when B is an irrational tori, since in this case T B = B × R ([CW16, combining Examples 3.23 and 4.19(3), and Theorem 4.15]) and the D -topology on B is not Hausdorff.Moreover, via Theorem 4.22 and Section 4.2, we get many examples of projective diffeo-logical vector spaces from classical differential geometry! References [CSW] J.D. Christensen, G. Sinnamon and E. Wu,
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