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: d A :! 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 Cartesian Restriction Tangent Differential Differential Differential Categories Categories Categories 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: Where we are Tensor Cartesian Restriction Tangent Integral Integral Integral Categories Categories Categories Categories Story Today 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: J A = ∆ ⊗ (1 ⊗ ǫ A )d A 1 + 1 A J A :! A → ! A 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). 1 Composition 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: s A :! A → ! A ⊗ A which axiomatizes: Polynomial integration On certain objects the Rota-Baxter and U -substitution rule. Dual to d A :! 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 object 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: Linear 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 with an additive system of linear maps is: Linear 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 15 / 36
System of Linear Maps A system of linear maps L on a category X with finite products is a subfibration of 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 ( v 1 , ..., c 1 v i + c 2 w i , ..., v n ) = c 1 f ( v 1 , ..., v i , ..., v n ) + c 2 f ( v 1 , ..., w i , ..., v n ) 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 × A 1 × ... × A n → B is n -linear in its last n arguments: g ( z ) · x 1 · ... · x n JS Lemay Work with: Robin Cockett, Richard Blute, Kristine Bauer ( University of Calgary) Cartesian Integral Categories August 12, 2016 17 / 36
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