Differential forms in non-linear Cartesian differential categories - - PowerPoint PPT Presentation

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Differential forms in non-linear Cartesian differential categories - - PowerPoint PPT Presentation

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Overview Cartesian differential categories Examples of Non-linear CDCS Differential forms Cohomology Non-linear tangent categories Differential forms in non-linear Cartesian differential categories Hayley Reid and Jonathan Bradet-Legris

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Overview Cartesian differential categories Examples of Non-linear CDCS Differential forms Cohomology Non-linear tangent categories

Differential forms in non-linear Cartesian differential categories

Hayley Reid and Jonathan Bradet-Legris Mount Allison University (joint work with Dr. Geoff Cruttwell) FMCS 2018 Sackville, Canada, June 1, 2018

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Overview Cartesian differential categories Examples of Non-linear CDCS Differential forms Cohomology Non-linear tangent categories

Overview

Cartesian differential categories (CDCs) Non-linear CDCs

Motivation for non-linear CDCs Examples Constructions on categories

Differential Forms

New definition for non-linear CDCs

Cohomology

Examples

Non-linear tangent categories

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Cartesian differential category definition

Definition (Blute, Cockett, Seely, 2009) A Cartesian differential category is a left additive category with chosen products which has, for each map f : X → Y , a map D(f ) : X × X − → Y such that: [CD.1] D(f + g) = D(f ) + D(g) and D(0) = 0; [CD.2] a + b, cD(f ) = a, cD(f ) + b, cD(f ) and 0, aD(f ) = 0; [CD.3] D(π0) = π0π0, D(π1) = π0π1, and D(1) = π0; [CD.4] D(f , g) = D(f ), D(g); [CD.5] D(fg) = D(f ), π1f D(g) (“Chain rule”); [CD.6] a, 0, 0, dD(D(f )) = a, dD(f ); [CD.7] a, b, c, dD(D(f )) = a, c, b, dD(D(f )).

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Examples of CDCs

Example Smooth is a CDC, with v, xD(f ) = [J(f (x))] · v, where [J(f )] is the Jacobian of f. Poly is a CDC, with the same derivative

Poly has the Rns as objects and polynomial functions as arrows

The category of abelian groups with group homomorphisms as arrows, with v, xD(f ) = f (v).

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The forward difference operator

The category abfun (objects: abelian groups, arrows: functions) with v, xD(f ) = f (x + v) − f (x) is not an example of a CDC. It satisfies every axiom except for the first part of [CD.2]. What kind of structure do we get if we simply remove the first part

  • f [CD.2]?
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Non-linear Cartesian differential category definition

Definition (Bradet-Legris, Cruttwell, Reid) A non-linear Cartesian differential category is a left additive category with chosen products which has, for each map f : X → Y , a map D(f ) : X × X − → Y such that: [NLCD.1] D(f + g) = D(f ) + D(g) and D(0) = 0; [NLCD.2] 0, aD(f ) = 0; [NLCD.3] D(π0) = π0π0, D(π1) = π0π1, and D(1) = π0; [NLCD.4] D(f , g) = D(f ), D(g); [NLCD.5] D(fg) = D(f ), π1f D(g); [NLCD.6] a, 0, 0, dD(D(f )) = a, dD(f ); [NLCD.7] a, b, c, dD(D(f )) = a, c, b, dD(D(f )).

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A few examples of Non-linear CDCs

Example All CDCs are non-linear CDCs. The category abfun, which has abelian groups as objects and functions as arrows, and the D arrow is v, xD(f ) = f (x + v) − f (x). Smooth, but changing D to be v, xD(f ) = f (x + v) − f (x).

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Simple slice categories

Definition (Blute, Cockett, Seely, 2009) For a category with products X and a fixed A ∈ X, the following structure is called a simple slice category, and is denoted X[A].

  • bjects: those of X

arrows: an arrow f from X to Y is an arrow f : X × A → Y composites: the composite X Y Z

f g

is X × A Y × A Z

π0f ,π1 g

identity: 1X : X × A X

π0

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Simple slice categories results

Theorem (From Blute, Cockett, Seely, 2009) Let C be a Cartesian differential category. Then C[A] is a Cartesian differential category, with D arrow DA(f ) = π0, 0, π1, π2D(f ), where D(f ) is the D arrow for C. Theorem Let C be a non-linear Cartesian differential category. Then C[A] is a non-linear Cartesian differential category, with D arrow DA(f ) = π0, 0, π1, π2D(f ), where D(f ) is the D arrow for C.

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Idempotent splitting categories

Definition An idempotent in a category is an arrow e : X → X such that ee = e. Definition The idempotent splitting category of a category C, denoted Idem(C) has

  • bjects: (X, eX) where eX is an idempotent on X.

arrows: an arrow f : (X, eX) → (Y , eY ) is an arrow f : X → Y such that the following diagram commutes X X Y Y

f eX f eY

identity: eX : (X, eX) → (X, eX) is the identity on (X, ex). composites: defined as in C.

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Idempotent splitting categories results

Definition In a Cartesian differential category, a map f is linear if D(f ) = π0f . Definition The linear idempotent splitting category of a Cartesian differential category C, denoted idemLin(C), is the full subcategory of idem(C) consisting of objects (X, e) such that e linear. Theorem Let C be a Cartesian differential category. Then idemLin(C) is a Cartesian differential category, with the same D arrow as C.

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Idempotent splitting category results

Definition The non-linear idempotent splitting category of a category C, denoted idemNLin(C), is the full subcategory of idem(C) consisting of

  • bjects (X,e) such that e is linear and additive.

Theorem Let C be a non-linear Cartesian differential category. The idemNLin(C) is a non-linear Cartesian differential category, with the same D arrow as C.

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Differential forms

Differential forms and exterior differentiation for CDCs were defined by Cruttwell in 2013. This definition required the use of the linearity condition ([CD.2]) to prove the naturality of the exterior derivative. We needed a new definition for differential forms and exterior differentiation for the non-linear CDCs.

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Important Definitions

Based on the definitions from [5] (i) Functor Qn : X → X (ii) Linear Objects (iii) Non-linear differential forms

− quasi-multilinear (preserves the 0 map) − skew-symmetric

(iv) Quasi exterior Derivative

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Q Functor and Linear Objects

  • Definition. Given a non-linear Cartesian differential category X, for any

n ≥ 1, there is an endofunctor Qn : X → X.

  • given an object M in X : Qn(M) = Q(M)n × M where

Q(M)n = M × M × . . . × M

  • n times
  • given a map f : M → M′ :

Qn(f ) = π0, 0D(f ), π1, 0D(f ), . . . , πn−1, 0D(f ), πnf

  • Definition. In a Non-Linear Cartesian Differential Category, say that

an object A is linear if Q(A) = A × A.

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Non-Linear Differential Forms

  • Definition. For any n ≤ 1 and 0 ≤ i ≤ n − 1, define the map

qi : L(M) × Qn(M) → Q(Qn(M)) by qi = 0, 0, . . . , 0, π0, 0, . . . , 0|π1, . . . , πi, 0, πi+2, . . . , πn+1 For a map f : TnM → A, say f is quasi-multilinear if for all 0 ≤ i ≤ n − 1 : M × Qn(M) Q(Qn(M)) Qn(M) A

qi π1,π2,...,πi,π0,πi+2,...,πn+1 D(f ) f

. Definition Say a map f is skew-symmetric if for any 0 ≤ i, j ≤ n − 1, the following is true : π0, . . . , πi, . . . , πj, . . . , πnf + π0, . . . , πj, . . . , πi, . . . , πnf = 0

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Non-Linear Differential Forms

Let X be a Non-Linear CDC. For an object M ∈ X, a linear object A ∈ X and n ≥ 1, define a non-linear differential n-form on M with values in A to be a map ω : Qn(M) → A which is quasi-multilinear and skew-symmetric. Denote the set of n-forms

  • n M with values in A by ΨA

n (M). Define ΨA 0 (M) to be the hom-set

X(M, A).

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Quasi Exterior Derivative

  • Definition. For n ≥ 1 and 0 ≤ i ≤ n − 1 and M an object, define the the

map ri to be M × Qn(M)

ri=0,...,0,πi|π0,..., ˆ πi,...,πn,0

− − − − − − − − − − − − − − − − − − − → Q(Qn(M)) where ˆ πi indicates the exclusion of that term. Suppose A is a linear group, and ω ∈ ΨA

n (M). For n ≥ 1, define the

quasi exterior derivative of ω, denoted ∂n(ω), to be the map given by ∂n(ω) :=

n

  • i=0

(−1)iriD(ω)

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Key Results

  • Let f : M′ → M, and ω : M → A be a non-linear differential n-form,

then Q(f )ω : M′ → A is also a non-linear differential n-form.

  • The quasi exterior derivative applied to a non-linear differential n-form

gives a non-linear differential (n + 1)-form.

  • The quasi exterior derivative is a natural transformation.
  • Applying the quasi exterior derivative twice to a non-linear differential

n-form gives the 0 map : ∂(∂(ω)) = 0.

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Key Results (1) & (2)

  • Lemma. Let f : M′ → M, and ω ∈ ΨA

n (M). Then the composite

Qn(M′)

Qn(f )

− − − → Qn(M)

ω

− − → A is in ΨA

n (M′).

(This allows us to view ΨA

n (−) as a functor from Xop to set.)

  • Lemma. For any ω ∈ ΨA

n (M), its exterior derivative ∂n(ω) is in

ΨA

n+1(M).

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Key Results (3) & (4)

  • Lemma. For any n ≥ 0 and differential group A, exterior differentiation

∂n : ΨA

n −

→ ΨA

n+1

is a natural transformation. ΨA

n (M)

ΨA

n (M′)

ΨA

n+1(M)

ΨA

n+1(M′) ΨA

n (f )

∂n(ω) ∂n(Qn(f )ω) ΨA

n+1(f )

  • Lemma. For any n ≥ 0 and linear group A, the following composition is

the 0 map : ΨA

n (−)

ΨA

n+1(−)

ΨA

n+2(−) ∂n ∂n+1

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Cohomology

  • The abelian groups ΨA

n (M) and quasi exterior derivatives ∂n for n ≤ 0

form a cochain complex:

{0} ΨA

0 (M)

ΨA

1 (M)

. . . ΨA

n (M)

. . . ∂0 ∂1 ∂n−1 ∂n

  • Call the cohomology groups of a cochain of this form the quasi De

Rahm cohomology of M. Let Hi

qdr(M, A) denote the ith quasi De Rahm cohomology group and

define ∂−1 := 0.

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Finding Specific Examples of Cohomology Groups

  • Lemma. For any pair of groups G, H, the first quasi De Rahm

cohomology group is H0

qdr(G, H) = H.

  • Proposition. The quasi De Rahm cohomology groups of non-linear

differential n-forms in abfun from Z2 to Z2:

{0} ΨZ2

0 (Z2)

ΨZ2

1 (Z2)

. . . ΨZ2

n (Z2)

. . . ∂0 ∂1 ∂n−1 ∂n

are all Z2 : Hqdr(Z2, Z2) = Z2, Z2 . . . Z2 . . .

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Integers and polynomial Functions

Consider the category of abelian groups with Zn’s as objects and polynomial functions as arrows {0} ΨZ

0(Z)

ΨZ

1(Z)

ΨZ

2(Z)

. . .

∂0 ∂1 ∂2

By the previous lemma : H0

qdr(Z, Z) = Z

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Integers and polynomial Functions

Consider the category of abelian groups with Zn’s as objects and polynomial functions as arrows {0} ΨZ

0(Z)

ΨZ

1(Z)

ΨZ

2(Z)

. . .

∂0 ∂1 ∂2

By the previous lemma : H0

qdr(Z, Z) = Z

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H1

qdr(Z, Z)

  • Found a basis for the kernel of ∂2 and image of ∂1
  • Kernel:

{v, xv, v 2, x2v + xv 2, v 3, x3v + xv 3, x2v 2, v 4, . . .}

  • Image:

{v, (2xv + 2v 2), (3x2v + 3xv 2 + v 3), . . . , v n}

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Non-Linear Tangent Categories

Can define a “non-linear” version of tangent categories such that any non-linear CDCs can be an example, with the following changes:

  • Remove the “+” natural transformation.
  • Additive bundles are replaced with pointed bundles
  • We require the following triple equalizer diagram to hold instead of an

equalizer for ℓ involving +: QM Q2M QM

ℓ

Q(p) p ppz

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Some structures in non-linear tangent categories

Connections

removed conditions involving “+” examples in abfun

Sector forms

same definition as regular tangent categories examples in smooth and abfun

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Future Work

  • More examples of non-linear CDCs
  • More cohomology examples in abfun.
  • Find out what Quasi De Rahm cohomology is in smooth and if it is

the same as the De Rahm cohomology.

  • Hqdr in idempotent and simple slice of non-linear CDCs.
  • More sector form calculations
  • More connections examples
  • Applications for non-linear differential structure
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References

References: [1] Blute, R. F., Cockett, J. R. B., and Seely, R. A. G. Cartesian Differential Categories, Theory

  • Appl. Cat. 22 (2009), no. 23, 662–672.

[2] Cockett, J. R. B., and Cruttwell, G. S. H. Differential structure, tangent structure, and SDG, Appl Categor Struct, 22 (2014) 331–417. [3] J. R. B. Cockett and G. S. H. Cruttwell. Connections in Tangent Categories. Theory and Applications of Categories, Vol. 32, No. 26, pg. 835-888, 2017. [4] Cruttwell, G. S. H., and Lucyshyn-Wright, R. B. B. A Simplicial Foundation for Differential and Sector Forms in Tangent Categories 2017, preprint. [5] G. S. H. Cruttwell, Forms and Exterior Differentiation in Cartesian Differential Categories, Theory Appl. Cat. 28 (2013), no. 28, 981–1001. [6] Hatcher, A. Algebraic Topology. Cambridge; Cambridge University Press, 2002; ISBN 0-521-79540-0. [7] Mendes, A.; Remmel, J. Counting with Symmetric Functions; Developments in Mathematics; Springer International Publishing, 2015; ISBN 978-3-319-23617-9.