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Cartesian Integral Categories CT 2016

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer

University of Calgary

August 12, 2016

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 1 / 36

How our story begins...

In the early 2000s, Ehrhard and Regnier introduced the notion of differentiation in linear logic with the differential λ-calculus and differential proof nets. !A ⊢ B !A, A ⊢ B

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 2 / 36

Birth of Categorical Differentiation

In 2006, Blute, Cockett and Seely introduced tensor differential categories: an additive symmetric monoidal category with a comonad (!, δ, ε) which is an coalgebra modality and a natural transformation: dA :!A ⊗ A →!A which axiomatizes the basic notions of differentiation: Differentiating constant maps; The product/Leibniz rule; Differentiating linear maps; The chain rule.

This part of the story doesn’t stop here...

Lots of of cool work has been done with tensor differential categories by Richard Blute, Rory Lucyshyn-Wright, Keith O’Neill, Thomas Ehrhard, Marcelo Fiore and many others.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 3 / 36

Rise of Categorical Differentiation

In 2008, Blute, Cockett and Seely introduced Cartesian differential categories: a Cartesian left additive category with a differential combinator: f : A → B D[f ] : A × A → B Linear which axiomatizes basic notions of directional derivation such as: Additivity of the differential combinator; Linearity of the differential combinator; The chain rule; Symmetry of the mixed partial derivatives.

Theorem

The coKleisli category of a tensor differential category (which is also a Seely category) is a Cartesian differential category.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 4 / 36

Crusade into Differential Geometry

In 2010, Cockett, Cruttwell and Gallagher introduced differential restriction categories: a Cartesian left additive restriction category with a differential combinator which axiomatizes the theory of partial differentiation. In 1984, Rosicky introduced an abstract structure associated of tangent bundle functors in algebraic and differential geometry. In 2013, Cockett and Cruttwell introduced tangent categories (inspired by and a slight generalization of Rosicky’s work) which axiomatizes tangent structure and the theory of smooth manifolds.

Theorem

The manifold completion of a differential restriction category with joins is a tangent category.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 5 / 36

Evangelism of Categorical Differentiation

This give us a story from elementary differentiation to differential geometry.

Tensor Differential Categories Cartesian Differential Categories Restriction Differential Categories Tangent Categories

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 6 / 36

On the road to Integral Enlightenment

Following the differential story, we are trying to get the integral story:

Tensor Integral Categories Cartesian Integral Categories Restriction Integral Categories Tangent Categories

Story Today Where we are

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 7 / 36

The story of integration so far...Ehrhard’s Work

In 2014 (published in 2016), Ehrhard defined the natural transformation: JA :!A →!A JA = ∆⊗(1 ⊗ ǫA)dA1 + 1A which all tensor differential categories have. Ehrhard defined that a tensor differential category had antiderivatives if J was a natural isomorphism. Which is a property and NOT structure! Antiderivatives lead to integration and the fundamental theorems of calculus in a tensor differential category. However, Ehrhard did not isolate integration (no differentiation involved).

1Composition is written diagramaticaly throughout this presentation.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 8 / 36

The story of integration so far...Tensor Integration

In 2015, Blute, Bauer, Cockett and Lemay introduced tensor integral categories: an additive symmetric monoidal category with a comonad (!, δ, ε) which is a coalgebra modality and natural transformation: sA :!A →!A ⊗ A which axiomatizes: Polynomial integration On certain objects the Rota-Baxter and U-substitution rule. Dual to dA :!A ⊗ A →!A in a tensor differential category. Compatibility between s and d axiomatize the fundamental theorems of calculus. Ehrhard’s notion of antiderivatives recaptures this structure of tensor integration and comaptibility.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 9 / 36

What does the story of tensor integration tell us?

The integral structure should be ”dual” to the differential strucutre; Polynomial Integration is the most fundamental notion of integration; Classical (volume and measure) integration is only recaptured on certain

• bject and not on the entire category. For example, in the category of vector

spaces, only integrating maps of the form f : V → R gives back the notion of volume/measure/area. Tensor integration can be obtained from tensor differentiation with ONE extra property. GOAL: The fundamental theorems of calculus.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 10 / 36

Constructing A Cartesian Integral Combinator

Now we will construct an integral combinator dual to the differential combinator: f : A → B D[f ] : A × A → B Linear

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 11 / 36

Cartesian Integral Combinator

An integral combinator S f : A × A → B S[f ] : A → B

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 12 / 36

Cartesian Integral Combinator

An integral combinator S on a Cartesian Left Additive Category is: f : A × A → B S[f ] : A → B

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 12 / 36

Cartesian Left Additive Category

A category X with finite products is a Cartesian left additive category if each hom-set is a commutative monoid, that is, we can add parallel maps f + g and there are zero maps 0, such that: f + g = g + f f + 0 = f = 0 + f And composition on the left preserves the additive structure, that is: f (g + h) = fg + fh f 0 = 0 And the product structure preserves the additive structure.

Example

The category of commutative monoids and set functions; The category of topological vector spaces and continuous functions.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 13 / 36

Cartesian Integral Combinator

An integral combinator S on a Cartesian Left Additive Category is: f : A × A → B S[f ] : A → B

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 14 / 36

Cartesian Integral Combinator

An integral combinator S on a Cartesian Left Additive Category is: f : A × A → B S[f ] : A → B Linear

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 14 / 36

Cartesian Integral Combinator

An integral combinator S on a Cartesian Left Additive Category with an additive system of linear maps is: f : A × A → B S[f ] : A → B Linear

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 15 / 36

System of Linear Maps

A system of linear maps L on a category X with finite products is a subfibration

• f the simple fibration which is closed under the additive structure.

Example

The category of topological vector spaces and continuous functions has an additive system of linear maps consisting of functions which are linear in certain arguments: f (v1, ..., c1vi + c2wi, ..., vn) = c1f (v1, ..., vi, ..., vn) + c2f (v1, ..., wi, ..., vn)

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 16 / 36

System of Linear Maps - Proto-Term Logic

To make our lives easier, for functions we use the · notation for linearity: If f : C × A → B is linear in its second argument: f (x) · y If g : C × A1 × ... × An → B is n-linear in its last n arguments: g(z) · x1 · ... · xn

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 17 / 36

Cartesian Integral Combinator

An integral combinator S on a Cartesian Left Additive Category with an additive system of linear maps is: f : A × A → B S[f ] : A → B Linear x : A, y : A → f (x) · y : B a : A → S[f ](a) : B

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 18 / 36

Cartesian Integral Combinator

An integral combinator S on a Cartesian Left Additive Category with an additive system of linear maps is: f : C × A × A → B SC[f ] : C × A → B Linear z : C, x : A, y : A → f (z, x) · y : B c : C, a : A → SC[f ](c, a) : B

Remark

For Cartesian differential categories, context differentiation comes for free: f : A → B D[f ] : A × A → B ⇒ g : C × A → B DC[g] : C × A × A → B

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 18 / 36

Proto-Integral Term Logic

We introduce the proto-integral term logic notation: SC[f ](c, a) = (

• x·y

f (c, x) · y)(a)

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 19 / 36

Proto-Integral Term Logic

We introduce the proto-integral term logic notation: SC[f ](c, a) = (

• x·dx

f (c, x) · dx)(a) However dx is simply notation and is not related to x.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 19 / 36

Proto-Integral Term Logic

We introduce the proto-integral term logic notation: SC[f ](c, a) = (

• x·dx

f (c, x) · dx)(a) However dx is simply notation and is not related to x. The integral SC[f ] should be thought of as the integral centred at zero: (

• x·dx

f (c, x) · dx)(a) =

a

• f (c, x)dx

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 19 / 36

Integral Combinator

An integral combinator S on a Cartesian Left Additive Category with an additive system of linear maps is: f : C × A × A → B SC[f ] : C × A → B Linear z : C, x : A, y : A → f (z, x) · y : B c : C, a : A → (

• x·dx

f (c, x) · dx)(a) : B such that S satisfies the following five properties:

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 20 / 36

Cartesian Integral Combinator Axioms

[S.1] SC[f + g] = SC[f ] + SC[g] and SC[0] = 0; [S.2] If h, k ∈ L[C] and and if the square on the left commutes, then the square

• n the right commutes:

C × A × A

π0,f

• π0,(1C ×π0)h,(1C ×π1)h
• C × B

k

• C × A′ × A′

g

B′ ⇒ C × A

π0,SC [f ]

• π0,h
• C × B

k

• C × A

SC [g]

B′ [S.3] g ∗(SD[f ]) = SC[g ∗(f )] [S.4] If f : C × A × B × A × B is bilinear then: SC×A[(1C × σA,B×B)SC×B×B[(1C × σB×B,A×A)(1C × 1A × σA,B × 1B)f ]] = (1C × σA,B)SC×B[(1C × σB,A×A)SC×A×A[(1C × 1A × σA,B × 1B)f ]] [S.5] If f : C × A×n → B is a map which is n-linear in its last n arguments then: n · SC[(1C × ∆n−1 × 1A)f ] = (1C × ∆n)f

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 21 / 36

Cartesian Integral Combinator Axioms

[S.1] Additivity:

(

• x·dx

f (c, x) · dx + g(c, x) · dx)(a) = (

• x·dx

f (c, x) · dx)(a) + (

• x·dx

g(c, x) · dx)(a) (

• x·dx

0)(a) = 0

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 22 / 36

Cartesian Integral Combinator Axioms

[S.2] is equivalent to the following two axioms: [S.2.a] Right Linear Substitution: (

• x·dx

g(c, h(c) · x) · (h(c) · dx))(a) = (

• x·dx

g(c, x) · dx)(h(c) · a) [S.2.b] Left Linear Substitution: (

• x·dx

k(c) · [f (c, x) · dx])(a) = k(c) · [(

• x·dx

f (c, x) · dx)(a)]

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 23 / 36

Cartesian Integral Combinator Axioms

[S.3] Context Substitution: (

• x·dx

(f (c, x) · dx)[g(d)/c])(a) = (

• x·dx

f (c, x) · dx)(a)[g(d)/c]

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 24 / 36

Cartesian Integral Combinator Axioms

[S.4] Fubini’s Theorem: If f : C × A × B × A × B → D is bilinear in its last two arguments: z : C, x : A, y : B, dx : A ; dy : B → f (z, x, y) · dx · dy : D then: (

• y·dy

(

• x·dx

f (c, x, y) · dx · dy)(a))(b) = (

• x·dx

(

• y·dy

f (c, x, y) · dx · dy)(b))(a)

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 25 / 36

Cartesian Integral Combinator Axioms

[S.5] Polynomial Integration: If f : C × A×n → D is n-linear in its last n arguments: z : C, x1 : A ; .... ; xn : A → f (z) · x1 · ... · xn then: n · (

• x·dx

f (c) · x · x · ... · x · dx)(a) = f (c) · a · a · ... · a [S.5] gives an infinite family of axioms for one variable polynomial integration: x xn dx = 1 n + 1xn+1

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 26 / 36

Cartesian Integral Category

A Cartesian integral category is a Cartesian left additive category with an additive system of linear maps and an integral combinator.

Proposition

The coKleisli category of a tensor integral category (which is also a Seely category) is a Cartesian integral category. Why ”proto-term logic”? In fact there is a term logic for integration:

Proposition

The integral term logic (NOT TRIVIAL!) is sound and complete.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 27 / 36

Example - Category of Entire Functions

Let ENTR be the category of entire functions defined as follows: [Objects]: Rn for each n ∈ N; [Maps]: Entire functions f : Rn → Rm, where f = f1, ..., fm is entire if every fk : Rn → R is entire. ENTR is a Cartesian left additive category. An entire function f : Rm × Rn → R is linear in Rn if: f (x1, .., xm) · (y1, ..., yn) = f1(x1, ..., xm)y1 + ... + fn(x1, ..., xm)ym For some entire functions fi : Rm → R:

Example

f (x, y) · (z, w) = x7z + y 3z + xyw is linear in (z, w). g(x, y, z, w) = xz + y 3w 2 is not linear in (z, w).

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 28 / 36

Example - Integral Combinator

An integrable function f : C × Rn × Rn → R is of the form: f ( c, x1, ..., xn) · (y1, ..., yn) =

n

• j=1
• p

ki,j(

c)a

ki,jxk1,j 1

...xkn,j

m yj

where p

ki,j(

c) is a monomial. The integral of f is defined as follows: SC[f ]( c, x1, ..., xn) =

n

• j=1
• p

ki,j(

z)a

ki,j

1 1 + n

i=1 ki,j

xk1,j

1

...xkp,j+1

p

xkn,j

m

The integral of an integrable map f : C × Rn × Rn → Rk is defined coordinate wise as: SC[f ] = SC[f1], ..., SC[fk]

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 29 / 36

Example - Polynomial Example

Example

Consider the function f : R2 × R2 → R defined as: f (x1, x2) · (y1, y2) = x2

1y2 + x3 1y1 + x5 2y2

Then the integral of f is: S[f ](a, b) = (

• (x,y)·(dx,dy)

x2dy + x3dx + y 5dy)(a, b) = 1 3a2b + 1 4a3 + 1 6b6

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 30 / 36

Example - Exponential Function

Example

Consider your favourite entire function ex = ∞

n=0 xn n! and define the function

f : R × R → R as: f (x) · y = exy =

∞

• n=0

xny n! Then the integral of f is: S[exy](a) = (

• x·dx

exdx)(a) = (

• x·dx

∞

• n=0

xn n! dx)(a) =

∞

• n=0

an+1 (n + 1)n! = ea − 1 = a exdx

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 31 / 36

Integration from Differentiation

We’ve given a seperate axiomatization of integration but polynomial integration give us an infinite list of axioms... However a Cartesian differential category with ONE extra property is an integral category. This extra property is called ”having antiderivatives”.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 32 / 36

Brief look at Differentiation

f : C × A → B DC[f ] : C × A × A → B Linear Cartesian differential categories also have a proto-term logic: DC[f ](c, a, b) = ∂f (c, x) ∂x (a) · b ”The partial derivative of f in x evaluated at a in direction b”.

Proposition

The differential term logic is sound and complete.

Proposition

Every Cartesian differential category has an additive system of linear maps: DC[f ](c, a, b) = ∂f (c) · x ∂x (a) · b = f (c) · b

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 33 / 36

Brief look at Antiderivatives

Recall that in the tensor side of the story, having antiderivative was the property that a certain natural transformation: JA :!A →!A which we always had, was an a natural isomorphism. So what does this correspond to in the Cartesian world?

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 34 / 36

Brief look at Antiderivatives

The J-Combinator is defined as: f : C × A × A → B JC[f ] : C × A × A → B Linear Linear JC[f ](c, a, b) = ∂f (c, x).b ∂x (a) · a + f (c, a) · b

Example

J[xny](a, b) = nan−1ba + anb = (n + 1)anb; J[exy](a, b) = eaba + eab = (a + 1)eab

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 34 / 36

Brief look at Antiderivatives

A Cartesian differential category has antiderivatives if the J-combinator has an inverse combinator: J−1

C [JC[f ]] = f = JC[J−1 C [f ]]

Proposition

A cartesian differential category which has antiderivatives is an integral category: SC[f ](c, a) = J−1

C [f ](c, a, a)

Example

Most examples of Cartesian Integral categories arise from antiderivatives; ENTR has antiderivatives.

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 34 / 36

Concluding Remarks

We’ve given a story of Cartesian integral categories. This story has a term logic and relates back to the tensor story by the coKleisli category. We’ve also seen how Cartesian differential categories give rise to an integral structure. Cartesian integral and differential categories together also give a story of the fundamental theorems of calculus. The next step is to work on Integral Restriction categories.

Tensor Integral Categories Cartesian Integral Categories Restriction Integral Categories Tangent Categories

JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 35 / 36

Thank You

END. Thank you!

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