Weil spaces and closed tangent structure June 2, 2018 1 / 30 - PowerPoint PPT Presentation

Weil spaces and closed tangent structure June 2, 2018 1 / 30

Overview W 1 -actegories ˚ Tangent categories 1 W 1 -presheaf cats ˚ 2 Embedding theorem Closed and representable tangent categories 3 2 / 30

Tangent categories and Weil algebras ´ a la Leung Tangent categories have a tangent functor T ´ ´ Ñ X X for which TM is thought of as the object of vectors tangent to M . There are also pullbacks of T T n ´ ´ Ñ X X for which T n M is thought of n -vectors anchored at the same point of M . 3 / 30

Tangent categories and Weil algebra ´ a la Leung Of course the tangent functor can be iterated T n ´ ´ ´ Ñ X X for which T n M is thought of as the n -dimensional singular microcubes anchored at a point of M . The pullbacks may also be iterated T n k T n 1 ´ ´ ´ Ñ X ¨ ¨ ¨ X ´ ´ ´ Ñ X X for which T n k p¨ ¨ ¨ T n 1 M ¨ ¨ ¨ q is thought of as ... 4 / 30

Tangent categories and Weil algebras ´ a la Leung Nodes ” tangent functor Edges ” pullback along p . Graph Tangent categories t 1 TM t 1 t 2 T 2 M t 1 t 2 T 3 M t 3 t 1 T 2 p TM q t 3 t 2 5 / 30

Tangent categories and Weil algebras ´ a la Leung ¿ What about? t 1 t 2 t 3 Here one can add in the t 1 – t 2 part, or the t 2 – t 3 part, but not the t 1 – t 3 part. 6 / 30

Tangent categories and Weil algebras ´ a la Leung The graphs that allow defining valid tangent structure are in correspondence with Weil 1 algebras . 7 / 30

¿ What are Weil 1 algebras? Let R be a ring (or cancellative rig, like N ). Form the Lawvere theory out of ¨ ¨ W n : “ R r x 1 , . . . , x n s{p x i x j q 1 ď i ď j ď n W 4 : ¨ ¨ and augmented maps. Weil 1 algebras, or W 1 is the closure of the above to R r x 1 , . . . , x n s{ I b R r y 1 , . . . , y m s{ J : “ R r x 1 , . . . , y m s{p I Y J q in augmented R algebras. G 1 b G 2 : G 1 G 2 8 / 30

Transvere limits in W 1 Proposition (Leung) W 1 is an FL theory. In W 1 the following are limits: 1. W n is the n fold product of R r x s{p x 2 q ; 2. If lim i V i is a limit, then for any A A b lim i V i » lim i A b V i ; 3. The following square a ` bx ` cy a ` bx ` cyz a ` bx ` cxz a a ` 0 x is a pullback Transverse limits: Limits constructed starting with 1 or 3, and inductively applying 2 are called transverse. 9 / 30

Tangent category Tangent structure [Rosick´ y, Cockett-Cruttwell,Leung] A tangent structure on X is exactly a strong (iso) monoidal functor W 1 ´ Ñ r X , X s that sends transverse limits to pointwise limits. Observation Tangent structures on X are in 1 – 1 correspondence with models of W 1 in r X , X s regarded as a limit sketch. 10 / 30

Weil prolongation as an actegory Uncurrying Leung’s theorem, a tangent structure is exactly an actegory ∝ X ˆ W 1 ´ ´ Ñ X so natural isomorphisms A ∝ p U b V q » p A ∝ U q ∝ V A ∝ R » A such that for every transverse limit lim i V i in W 1 A ∝ p lim i V i q » lim i p A ∝ V i q For example TM “ M ∝ R r x s{p x 2 q T 2 M “ M ∝ R r x , y s{p x 2 , xy , y 2 q This means W 1 is a tangent category: ∝ ” b . 11 / 30

Functor categories involving a tangent category When Y is a tangent category then r X , Y s is too. F ∝ U : X ´ Ñ Y M ÞÑ F p M q ∝ U When X is a tangent category then r X , Y s is not. ... but should be? F ∝ U : X ´ Ñ Y M ÞÑ F p M ∝ U q The microlinear functors do (definitionally) form a tangent category. 12 / 30

Microlinear Weil spaces The category of Weil spaces [Bertram ‘14] is r W 1 , Set s The category of microlinear Weil spaces then forms a tangent category, M- W 1 . Our goal is to understand and then exploit M- W 1 . 13 / 30

Sketches of 1 {8 W 1 is a finite limit sketch under transverse limits. Proposition For a finite limit sketch, Mod p T q is a locally finitely presentable category. Observation M - W 1 is locally finitely presentable hence complete and cocomplete. 14 / 30

Sketches of 1 {8 Proposition (Kennison 1968) M - W 1 is a reflective subcategory of Psh p T q containing all W 1 p U , q . This is a move that will help us to characterize M- W 1 enriched categories. 15 / 30

M- W 1 is self enriched Observation Y p U b V q » Y p U q ˆ Y p V q Proposition For any Weil space, M ∝ U » r Y p U q , M s . r Y p U q , M sp X q » Nat p Y p U q ˆ Y p X q , M q » Nat p Y p U b X q , M q » M p U b X q ” p M ∝ U qp X q 16 / 30

M- W 1 is self enriched Proposition When M is microlinear, then for any Weil space X, r X , M s is microlinear. Hence M - W 1 is an exponential ideal. r X , M s ∝ lim i V i » r lim i V i , r X , M ss » r X , M ∝ lim i V i s » lim i pr X , M s ∝ V i q Corollary L The reflector Psh p W 1 q Ñ M - W 1 preserves products, and the ´ ´ product functor M ˆ is cocontinuous. Corollary M - W 1 is a monoidally reflective subcategory of Psh p W 1 q . 17 / 30

M- W 1 is self enriched Proposition M - W 1 categories form a reflective subcategory of Psh p W 1 q categories, and similarly for those categories that admit powers by representables. Corollary Let X be a Psh p W 1 q enriched category, such that for each A , B, X p A , B qp q is in M - W 1 , then X is in fact a M - W 1 enriched category. 18 / 30

Enriched Characterization of Tangent Structure When X is a tangent category, every homset gives rise to a functor X p A , B qp q : “ X p A , B ∝ q : W 1 ´ Ñ Set This functor preserves transverse limits. Proposition (Garner) A tangent category is exactly a category enriched in M - W 1 that admits powers by representable functors. A & Y p U q “ A ∝ U 19 / 30

M- W 1 has self enriched limits Proposition M - W 1 is complete and cocomplete as an M - W 1 category. This follows from Kelly chapter 3. M- W 1 is enriched in M- W 1 and has limits, colimits, and copowers. We use this to setup enriched limits and and colimits on enriched presheaf categories, and to characterize enriched natural transformations. 20 / 30

M- W 1 is a model of SDG part 1 Proposition The tangent functor is representable: that is TM » r D , M s for some D. Proof. Take D “ Y p R r x s{p x 2 qq . From a previous result, M ∝ U » r Y p U q , M s for all U , and hence TM » r Y p R r x s{ x 2 q , M s 21 / 30

M- W 1 is a model of SDG part 2 Proposition M - W 1 has a ring of line type, that satisfies the Kock-Lawvere axiom. In a complete, representable tangent category, one may form the ring of homotheties mirroring a move of Kol´ aˇ r. In SMan homotheties of the tangnet bundle is up to isomorphism R . 22 / 30

M- W 1 is a model of SDG part 2 Proposition M - W 1 has a ring of line type, that satisfies the Kock-Lawvere axiom. Start with r D , D s R 1 { p 1 D 0 Using equalizers obtain a subobject, which is a ring 22 / 30

Embedding theorem for tangent categories Proposition (Garner) Let X be a tangent category. The M - W 1 -category r X op , M - W 1 s is a representable tangent category. Corollary (Garner) Every tangent category embeds into a representable tangent category. 23 / 30

Transverse limits in a tangent category A transverse limit in a tangent category is a lim i A i such that p lim i A i q ∝ V » lim i p A i ∝ V q Transverse limits include: P Products A ˆ B ∝ W » p A ∝ W q ˆ p B ∝ W q ; P Pullback powers of p : A ∝ W 2 A ∝ W { p A ∝ W A p 24 / 30

Transverse limits in a tangent category When X is a tangent category and lim i B i is transverse, then for each A P X and U P W 1 : X p A , p lim i B i q ∝ U q » X p A , lim i p B i ∝ U qq » lim i X p A , B i ∝ U q Hence X p A , lim i B i qp q » lim i X p A , B i qp q : W 1 ´ Ñ Set is an enriched limit. Observation The enriched Yoneda embedding B ÞÑ X p , B qp q sends transverse limits to limits. 25 / 30

Closed tangent categories A cartesian closed tangent category is coherently closed when the induced map ψ θ ev ∝ U A ˆ r A , B s ∝ U ´ ´ Ñ p A ˆ r A , B sq ∝ U ´ ´ ´ ´ ´ Ñ B ∝ U r A , B s ∝ U ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ Ñ r A , B ∝ U s ψ “ λ p θ ; ev ∝ U q is invertible for each U P W 1 . So that coherently closed 26 / 30

Closed tangent categories via M- W 1 Proposition When X is a coherently closed tangent category, then for each A, the M - W 1 -functor A ˆ has a M - W 1 -right adjoint r A , s . Corollary The Yoneda embedding A ÞÑ X p , A qp q preserves the internal hom. That is for each U, X p , r A , B sqp U q » r X p , A qp U q , X p , B qp U qs 27 / 30

Endomorphisms of the tangent functor In a complete coherently closed tangent category, the endomorphisms of the tangent bundle may be computed as ż ż End p T q : “ X p Tc , Tc q ” r Tc , Tc s c c We also have the subobject of tangent transformations: Tan p T q ֌ End p T q This can be built in stages by equalizers taking those transformations which are commutative additive bundle homomorphisms that commute with the lift. In a representable tangent category, the tangent transformations can be simplified. Moreover, they always give the ring of line type. 28 / 30

Endomorphisms of the tangent functor In a complete coherently closed tangent category, the endomorphisms of the tangent bundle may be computed as ż ż End p T q : “ X p Tc , Tc q ” r Tc , Tc s c c We also have the subobject of tangent transformations: Tan p T q ֌ End p T q E.g. r T , p s Hom p T q r T , T s r T , 1 s p 1 etc 28 / 30