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:

◮ an endofunctor X T

− − → X;

◮ a natural transformation T p

− − → I;

◮ for each M, the pullback of n copies of T(M) pM

− − − → M along itself exists (and is preserved by T), call this pullback Tn(M);

◮ such that for each M ∈ X, T(M) pM

− − − → M has the structure

• f a commutative monoid in the slice category X/M, in

particular there are natural transformation T2

+

− − → T, I − − → 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 c2 = 1;

◮ (vertical lift) there is a natural transformation ℓ : T −

→ T 2 which preserves additive bundle structure and satisfies ℓc = ℓ;

◮ various other coherence equations for ℓ and c; ◮ (universality of vertical lift) the following is a pullback

diagram: T2(M)

π0p=π1p

• ν:=π0ℓ,π10T T(+) T 2(M)

T(p)

• M

T(M)

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 , π1f . (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: 1

s0

− − → 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 s0 is the unique curve leaving s0 which follows the vector field. That is c : (a, b) − → M, with (a, b) maximal, and 0 ∈ (a, b) so that 1

s0

• ❇

❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇

(a, b)

c

• d

T((a, b))

T(c)

• M

X

T(M)

where d is the “standard” vector field on (a, b) and c is the unique morphism vector fields determined by s0 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 = 1M = ST(p), and a starting position and direction t0 : 1 − → T(M) and S : T(M) − → T 2(M) the “behaviour” of a geometric system is a geodesic starting with direction and position determined by t0: this is the unique curve from a maximal (a, b) with 0 ∈ (a, b) determined by: 1

t0

• ❈

❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈

(a, b)

d

• c′
• T((a, b))

T(c′)

• T(M)

S

T 2(M)

where the geodesic is c = c′p : (a, b) − → M which is determined by 0c = t0p 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 H : T(M) ×M T(M) = T2(M) − → T 2(M) and this map determines a “spray” S = δH : T(M) − → T 2(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:

E2

π0q=π1q

• ν:=π0λ,π10T(σ) T(E)

T(q)

• M

T(M)

where E2 is the pullback of q along itself;

◮

Connections in Tangent Categories Connections Differential bundles

Differential bundles

(i) The “trivial” differential bundle is 1M = (1M, 1M, 1M, 0M). (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

• ❘

❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘

E

q

• λ
• ♣

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

M

0,ζ

T(M) ×M E

◮ 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 = pM.

Connections in Tangent Categories Connections Horizontal and vertical connections

Horizontal connection

A horizontal connection is a horizontal lift so that T(M) ×M E

π0

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

H

T(E)

T(q)

• T(M)

T(M) ×M E

π1

• ▼

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

H

T(E)

p

• E

satisfying conditions requiring the horizontal lift be linear: T(M)×M E

ℓ×0

• H

T(E)

ℓ

• T(T(M)×M E) T(H)

T 2(E)

T(M)×M E

0×λ

• H

T(E)

T(λ)c

• T(T(M) ×M E) T(H)

T 2(E)

An affine horizontal connection is torsion free when Hc = π1, π0H

Connections in Tangent Categories Connections Horizontal and vertical connections

Vertical connection

A vertical connection is a vertical descent so that: E

λ

• ❖

❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖

T(E)

K

E

satisfying conditions requiring the vertical descent be linear: T(E)

ℓ

• K

E

λ

• T 2(E)

K

T(E)

T(E)

T(λ)

• K

E

λ

• T 2(E)

K

T(E)

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(pM) × 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 XpM = 1M and Y : M − → E has Yq = 1M and the result is a vector field as ∇K(X, Y )q = XT(Y )Kq = XT(Y )T(q)p = Xp = 1M This is thought of as a “derivative of Y along the vector field X”.

Connections in Tangent Categories Connections Horizontal and vertical connections

Connections

Definition

A connection on a differential bundle q : E − → M consists of:

◮ a vertical connection K; ◮ a horizontal connection H; ◮ such that HK = 0 and K, pν + T(q), pH = 1.

A connection on the tangent bundle p : T(M) − → M is called an affine connection.

Connections in Tangent Categories Connections Horizontal and vertical connections

Connections

Proposition

If a differential bundle q has a connection (K, H) then T(E) is the pullback (over M) of T(M) × E and E. That is the following is a pullback: T(E)

T(q),p K

• T(M) ×M E

π0p=π1q

• E

q

M

splitting T(E) into horizontal and vertical subspaces

Connections in Tangent Categories Connections Horizontal and vertical connections

Examples of connections

Any differential object, A, is equivalently a differential bundle over 1 and all differential objects have a unique connection defined by:

◮ K : T(A) −

→ A by K(v, a) := v and

◮ H : A −

→ T(A) by H(a) := (0, a).

Connections in Tangent Categories Connections Horizontal and vertical connections

Examples of connections

Any differential object, A, is equivalently a differential bundle over 1 and all differential objects have a unique connection defined by:

◮ K : T(A) −

→ A by K(v, a) := v and

◮ H : A −

→ T(A) by H(a) := (0, a). Applying the tangent functor to a differential object A gives the differential bundle p : T(A) − → A and a canonical (affine) connection:

◮ K ′ : T 2(A) −

→ T(A) by K(d, v, w, a) := (d, a) and

◮ H′ : T(A) ×A T(A) −

→ T 2(A) by H(v, w, a) := (0, v, w, a).

Connections in Tangent Categories Connections Horizontal and vertical connections

K from H

Proposition

Suppose H is a horizontal connection on a differential bundle q : E − → M then the pair ({1 − Tq, pH}, H) is a connection on q. Note that this requires negatives! It also uses the universal property of λ.

Connections in Tangent Categories Connections Horizontal and vertical connections

H from K

Proposition

Let K be a vertical connection on a differential bundle q : E − → M. If h is a horizontal lift, that is a section of T(q), p, then the pair (K, h(1 − (K, pλ)) is a connection on q. This also requires negatives, but also needs T(q), p to have a horizontal lift h (the resulting connection is independent of the choice of h).

Connections in Tangent Categories Sprays The definition

Definition

A spray is a map S : T(M) − → T 2(M) such that [Spray.1] S is a second order vector field so Sp = 1T(M) = ST(p); [Spray.2] S is linear (or homogenous) in the following sense that T(M)

ℓ

• S

T 2(M)

T(ℓ) T 3(M) ℓ

T 4(M)

T(c) T 4(M) ⊕

• T 2(M)

T(S)

T 3(M)

commutes, where ⊕ := T(p), p+; [Spray.3] δνT(S) = (ℓT(S) +2 Sℓ) +1 (ST(ℓ)c +2 S0); [Spray.4] 0, ℓT(ν)T 2(S) = (ℓT(S)0+3ST(ℓ)cT(ℓ))+2(ST(ℓ)ℓT(c)+3ℓT(S)T(0)). A spray which satisfies Sc = S is said to be torsion free.

Connections in Tangent Categories Sprays The definition

Tangent circuits for [Spray. 1]

S

• =

=

S

Connections in Tangent Categories Sprays The definition

Meaning of [Spray. 2]

Lemma

When S satisfies [Spray.1] and [Spray.2] then ℓT(S) = ℓT 2(0) +0 ℓT(0) and 0S = 00. In particular, ϑ := ℓT(S), does not depend on S! Here is the calculation: ℓT(S) = ST(ℓ)ℓT(c)⊕ = ST(ℓ)ℓT(c)T(p), p + = ST(ℓ)ℓT(c)T(p), ST(ℓ)ℓT(c)p + = ST(ℓ)ℓT 2(p), ST(ℓ)ℓpc + = ST(ℓp)ℓ, ST(ℓ)p0c + = ST(p0)ℓ, SpℓT(0) + = T(0)ℓ, ℓT(0)+ = ℓT 2(0), ℓT(0)+ Thus, ϑ := ℓT 2(0) +0 ℓT(0) : T(M) − → T 3(M).

Connections in Tangent Categories Sprays The definition

Meaning of [Spray. 2]:

S

• =

S

• ⊕

=: ϑ

The maps ϑ and ϑ′, below, will now become rather important!

ϑ′

:=

ϑ

Connections in Tangent Categories Sprays The definition

Meaning of [Spray. 2]

We have the following manipulations for these maps:

ϑ = ϑ ϑ

• =
• ϑ
• =

ϑ

Connections in Tangent Categories Sprays The definition

Meaning of [Spray.2]

ϑ

• =

ϑ′

• ϑ
• =

ϑ

• ⊕

ϑ ⊕ = ϑ ϑ ⊕ ⊕ ⊕

Connections in Tangent Categories Sprays The definition

Meaning of [Spray.3]

In the presence of negatives [Spray.3] admits a re-expression:

Proposition

In any tangent category with negatives, in the presence of [Spray.1] and [Spray.2], [Spray.3] is equivalent to the requirement: [δν, S] = S. Where δ = 1, 1 : T(M) − → T2(M) is the diagonal map into the fiber product and ν is from the universality of the lift. In fact, this is often how the definition is given classically.

Connections in Tangent Categories Sprays The definition

Meaning of [Spray. 4]

For torsion free sprays we may re-express [Spray.3] and [Spray.4] using the equation: ϑ′S = Sℓ(2)

ϑ′ S

=

S ϑ ϑ ⊕ ⊕

ℓ(2) will become important: ℓ(2) :=

ϑ ϑ ⊕ ⊕

Connections in Tangent Categories Sprays Limit assumptions

Limit assumptions

In order to obtain the Ambrose-Palais-Singer theorem we need to introduce two limit assumptions (both preserved by T n): (A) The equalizer W (M)

ι

− → T 2(M) T(p) − − − − − → − − − − − → p T(M) exists. (B) The following diagram is an equalizer: T 2(M)

u

− − → T 2(W (M)) u0 − − − → − − − → u1 T(W 2(M)) with u, u0, and u1 defined by: T 2(M)

u ℓ(2)

• ▲

▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

T(W (M))

T(ι)

• T 3(M)

T 2(W (M))

ui

• T(ι)
• T 2(W 2(M))

T(ι)T 3(ι)

• T 4(M)

u′

i

T 6(M)

where u′

0 = ℓ(2)T 2(M) and u′ 1 = T 2(ℓ(2)).

Connections in Tangent Categories Sprays Limit assumptions

On the these assumptions

Observations:

• 1. Any Cartesian differential category satisfies (A): this amount

to the fact that the diagonal map is the equalizer of the

• projections. Moreover, the diagonal also equalizes the

symmetry and the identity map. Thus, all sprays are torsion free.

• 2. In any model of SDG the microlinear spaces satisfy (A) and

(B). In fact, (B) is directly from Lavendhomme! In these settings all sprays are torsion free.

• 3. In classical differential geometry (A) and (B) are true and all

sprays are torsion free.

• 4. In a general tangent category neither (A) nor (B) need be true

nor (as far as I can see) need all sprays be torsion free!

Connections in Tangent Categories Sprays Limit assumptions

Sprays and (A) ..

Given a spray (or a second-order vector field) S we may factorize it: T(M)

σ

• S
• ❚

❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚

W (M)

ι

T 2(M)

T(p)

• p

T(M)

Notice that, when W (M)

ι

− → T 2(M) c − − − − − − → − − − − − − → 1T 2(M) T 2(M) is an equalizer as well, then all sprays are torsions free!

Connections in Tangent Categories Sprays The Ambrose-Palais-Singer theorem

The Ambrose-Palais-Singer theorem for tangent categories

Theorem

In any tangent category satisfying assumptions (A) and (B) there is a bijective correspondence between torsion free sprays and torsion free horizontal connections on an object. I am going to present a fragment of the proof using tangent circuits ...

Connections in Tangent Categories Sprays The Ambrose-Palais-Singer theorem

A key map and lemma

η := π0ℓT(0) +2 π1ℓ0 : T2(M) − → T 3(M)

Lemma

T2(M)

ηT 2(σ)

− − − − − → T 2(W (M)) u0 − − − → − − − → u1 T 2(W 2(M)) commutes, where σ is the unique factorization of S through ι. Then H can be defined by: T2(M)

H

• ηT 2(σ)
• ❘

❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘

T 2(M)

u

T 2(W (M))

It is then “straightforward” to show H is a horizontal connection.

Connections in Tangent Categories Sprays The Ambrose-Palais-Singer theorem

Key property of η ...

η ϑ′

=

η ϑ ϑ ⊕ ⊕

Connections in Tangent Categories Sprays The Ambrose-Palais-Singer theorem

The key step in the proof!

η ⊕ ⊕ ϑ ϑ s

=

η ϑ′ S

=

η S ϑ ϑ ⊕ ⊕

Connections in Tangent Categories Sprays The Ambrose-Palais-Singer theorem

Aftermath ..

◮ Essentially translating DG and SDG proofs into the setting of

tangent categories (not so easy: languages are far apart!).

◮ The result seem to be more general proofs ... but no killer

separating examples (yet!)

◮ Description of connections is very smooth in tangent

categories.

◮ Proofs using tangent circuits are very straightforward ...

... once one has the basics components organized! Sprays and the Ambrose-Palais-Singer theorem is another right of passage for tangent categories!

Connections in Tangent Categories Sprays The Ambrose-Palais-Singer theorem

References:

◮ Cockett, R. and Cruttwell, G. Differential structure, tangent

structure, and SDG. in Applied Categorical Structures Volume 22, Issue 2, pp 331–417 (2014)

◮ Cockett, R. and Cruttwell, G. Differential Bundles. (On ArXiv) ◮ Cockett, R. and Cruttwell, G. Connections in tangent

• categories. (Manuscript)

◮ Lavendhomme, R. Basic concepts of Synthetic Differential

• Geometry. Kluwer Academic Publishers (1996)

◮ Rosick´

y, J. Abstract tangent functors. Diagrammes, 12, Exp.

• No. 3 (1984)