Connections in Tangent Categories Robin Cockett University of - PowerPoint PPT Presentation

Connections in Tangent Categories Connections in Tangent Categories Robin Cockett University of Calgary (joint work with Geoff Cruttwell) Category Theory 2016 Dalhousie

Connections in Tangent Categories Tangent categories The basics Tangent categories Definition (Rosicky 1984, modified Cockett/Cruttwell 2013) A tangent category consists of a category X with: T ◮ an endofunctor X − − → X ; p ◮ a natural transformation T − − → I ; p M ◮ for each M , the pullback of n copies of T ( M ) − − − → M along itself exists (and is preserved by T ), call this pullback T n ( M ); p M ◮ such that for each M ∈ X , T ( M ) − − − → M has the structure of a commutative monoid in the slice category X / M , in + particular there are natural transformation T 2 − − → T , I 0 − − → T ;

� � Connections in Tangent Categories Tangent categories The basics Tangent category definition continued... Definition ◮ (canonical flip) there is a natural transformation c : T 2 − → T 2 which preserves additive bundle structure and satisfies c 2 = 1; → T 2 ◮ (vertical lift) there is a natural transformation ℓ : T − which preserves additive bundle structure and satisfies ℓ c = ℓ ; ◮ various other coherence equations for ℓ and c ; ◮ (universality of vertical lift) the following is a pullback diagram: ν := � π 0 ℓ,π 1 0 T � T (+) � T 2 ( M ) T 2 ( M ) π 0 p = π 1 p T ( p ) � T ( M ) M 0

Connections in Tangent Categories Tangent categories The basics Examples (i) Finite dimensional smooth manifolds with the usual tangent bundle structure. (ii) Any Cartesian differential category is a tangent category, with T ( A ) = A × A and T ( f ) = � Df , π 1 f � . (iii) The infinitesimally linear objects in any model of synthetic differential geometry (SDG).

Connections in Tangent Categories Tangent categories The basics Some theory of tangent categories (i) A vector field on M is a map X : M − → TM which is a section of p : TM − → M . (ii) In a tangent category with negatives, vector fields have a Lie bracket operation [ X , Y ] which satisfies the Jacobi identity. (iii) T is always a monad unit 0 : M − → T ( M ) multiplication ⊕ := � T ( p ) , p � + : T 2 ( M ) − → T ( M ).

� � � Connections in Tangent Categories Connections and sprays Motivation Dynamical systems A dynamical system is specified by a vector field and a start state: s 0 1 − − → M and X : M − → T ( M ) As X is a vector field it is a section of p : T ( M ) − → M . The “behaviour” of the dynamical system starting at s 0 is the unique curve leaving s 0 which follows the vector field. That is c : ( a , b ) − → M , with ( a , b ) maximal, and 0 ∈ ( a , b ) so that 0 � d � T (( a , b )) 1 ( a , b ) ❇ ❇ ❇ ❇ ❇ c T ( c ) ❇ s 0 ❇ ❇ ❇ � T ( M ) M X where d is the “standard” vector field on ( a , b ) and c is the unique morphism vector fields determined by s 0 and X . Note: the differential of c is c ′ := dT ( c ) so c ′ = cX .

� � � � Connections in Tangent Categories Connections and sprays Motivation Geometrical system A geometrical system (or second-order dynamical system) is specified by a second order vector field , S , so that Sp = 1 M = ST ( p ), and a starting position and direction → T 2 ( M ) t 0 : 1 − → T ( M ) and S : T ( M ) − the “behaviour” of a geometric system is a geodesic starting with direction and position determined by t 0 : this is the unique curve from a maximal ( a , b ) with 0 ∈ ( a , b ) determined by: 0 � 1 ( a , b ) T (( a , b )) ❈ d ❈ ❈ ❈ ❈ T ( c ′ ) c ′ ❈ ❈ t 0 ❈ ❈ � T 2 ( M ) T ( M ) S where the geodesic is c = c ′ p : ( a , b ) − → M which is determined by 0 c = t 0 p and by the differential of c being c ′ .

Connections in Tangent Categories Connections and sprays Motivation Geometries The fundamental theorem – and most important result – of Riemannian Geometry states: Theorem A Riemannian manifold determines a unique torsion free, metric compatible connection called the Levi-Civita connection. A connection may be specified by a map → T 2 ( M ) H : T ( M ) × M T ( M ) = T 2 ( M ) − → T 2 ( M ) and this map determines a “spray” S = δ H : T ( M ) − which, in particular, is a second order vector field. Classical geometries give Geometrical systems!

Connections in Tangent Categories Connections and sprays Motivation Sprays and connections There is a further crucial – but more technical – classical result called the Ambrose-Palais-Singer theorem, it states: Theorem Sprays are in bijective correspondence to torsion free connections. It is known to hold both classically and in all models of Synthetic Differential Geometry (SDG). Does Ambrose-Palais-Singer hold in tangent categories?

Connections in Tangent Categories Connections and sprays Motivation Sprays and connections NO!! Counterexample: any Cartesian differential category without “halving” (i.e. division by 2)! So the question becomes: When does Ambrose-Palais-Singer hold in tangent categories? We present an answer and a feeling for proofs in tangent categories!

Connections in Tangent Categories Connections and sprays Motivation Sprays and connections in tangent categories Connections have a very natural expression in tangent categories.

Connections in Tangent Categories Connections and sprays Motivation Sprays and connections in tangent categories Connections have a very natural expression in tangent categories. Sprays are more complicated!!

� � Connections in Tangent Categories Connections Differential bundles Differential bundles Definition A differential bundle in a tangent category consists of an additive bundle q : E − → M with a map λ : E − → T ( E ), called the lift , such that ◮ all pullbacks along q exist and are preserved by T ; ◮ ( λ, 0) and ( λ, ζ ) are additive bundle morphisms; ◮ the following is a pullback diagram: ν := � π 0 λ,π 1 0 � T ( σ ) � T ( E ) E 2 π 0 q = π 1 q T ( q ) � T ( M ) M 0 where E 2 is the pullback of q along itself; ◮

Connections in Tangent Categories Connections Differential bundles Differential bundles (i) The “trivial” differential bundle is 1 M = (1 M , 1 M , 1 M , 0 M ). (ii) The tangent bundle p : T ( M ) − → M is a differential bundle. (iii) A differential object A (analogue of a “vector space”) ( T ( A ) = A × A ) is precisely a differential bundle over the final object 1. Vector bundles of Differential Geometry are differential bundles. However, a differential bundle need not have fibres of fixed dimension so the converse is not true.

� � � Connections in Tangent Categories Connections Horizontal and vertical connections Two fundamental maps A differential bundle has two key maps involving T ( E ) whose composite is the zero map: T ( E ) ❘ ❘ ❘ ♣ ❘ � T ( q ) , p � ♣ ❘ λ ♣ ❘ ♣ ❘ ♣ ❘ ♣ ❘ ❘ ♣ ❘ ♣ ❘ ♣ ❘ ♣ ❘ ♣ ♣ � T ( M ) × M E E ♣ M q � 0 ,ζ � ◮ The horizontal descent is � T ( p ) , q � : T ( E ) − → T ( M ) × M E ◮ The vertical lift is λ : E − → T ( E ) ◮ A section of the horizontal descent is call a horizontal lift ◮ A retraction of the vertical lift is called a vertical descent These are affine when E = T ( M ) and q = p M .

� � � � � � � � Connections in Tangent Categories Connections Horizontal and vertical connections Horizontal connection A horizontal connection is a horizontal lift so that H � T ( E ) H � T ( E ) T ( M ) × M E T ( M ) × M E ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ p ▼ T ( q ) ▼ ▼ ▼ ▼ π 0 π 1 ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ E T ( M ) satisfying conditions requiring the horizontal lift be linear: H � T ( E ) H � T ( E ) T ( M ) × M E T ( M ) × M E T ( λ ) c ℓ × 0 ℓ 0 × λ � T 2 ( E ) � T 2 ( E ) T ( T ( M ) × M E ) T ( H ) T ( T ( M ) × M E ) T ( H ) An affine horizontal connection is torsion free when Hc = � π 1 , π 0 � H

� � � � � Connections in Tangent Categories Connections Horizontal and vertical connections Vertical connection A vertical connection is a vertical descent so that: E ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ λ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ � E ❖ T ( E ) K satisfying conditions requiring the vertical descent be linear: K K � E � E T ( E ) T ( E ) T ( λ ) ℓ λ λ � T ( E ) � T ( E ) T 2 ( E ) T 2 ( E ) K K An affine vertical connection is torsion free in case cK = K .

Connections in Tangent Categories Connections Horizontal and vertical connections Covariant derivative In texts connections are often defined as covariant derivatives, that is bilinear forms on vector fields: ∇ : χ ( p M ) × χ ( q ) − → χ ( q ) Given a vertical connection K : T ( E ) − → E we obtain a covariant derivative: ∇ ( X , Y ) = XT ( Y ) K here X : M − → T ( M ) has Xp M = 1 M and Y : M − → E has Yq = 1 M and the result is a vector field as ∇ K ( X , Y ) q = XT ( Y ) Kq = XT ( Y ) T ( q ) p = Xp = 1 M This is thought of as a “derivative of Y along the vector field X ”.