Tutorial: Differential Categories and Cartesian Differential - PowerPoint PPT Presentation

Tutorial: Differential Categories and Cartesian Differential Categories JS Pacaud Lemay FMCS 2019 Thanks for the invitation!

The Differential Category World: The Four Tomes Cartesian Differential Differential Categories Categories Blute, Cockett, Seely - 2006 Blute, Cockett, Seely - 2009 Restriction Tangent Differential Categories Categories Rosicky - 1984 Cockett, Cruttwell, Gallagher - 2011 Cockett, Cruttwell - 2014

The Differential Category World: A Taster Multivariable Differential Differential Euclidean Differential Calculus Algebraic Foundations of Linear Logic Spaces λ -calculus Differentiation Cartesian Differential Algebraic Differential Smooth Categories De Rham Categories Functions Cohomology Blute, Cockett, Seely - 2006 Blute, Cockett, Seely - 2009 K¨ ahler Directional Differential Differentials Derivative Algebras De Rham Cohomology Algebraic Geometry Open Subsets Tangent Categories Differential Tangent Restriction Geometry Bundles Rosicky - 1984 Partial Differential Smooth Cockett, Cruttwell - 2014 Categories Functions Cockett, Cruttwell, Gallagher - 2011 Vector SDG Bundles Smooth Manifolds

The Differential Category World: It’s all connected! Restriction Differential Categories s p a M Cockett, Cruttwell, Gallagher - 2011 l a t o T coKleisli ⊂ Manifold Completion Cartesian Differential Differential Categories Categories Blute, Cockett, Seely - 2006 Blute, Cockett, Seely - 2009 ⊗ -Representation ⊂ Differential Objects c o - E Tangent i l e n b e Categories r g - M o o Rosicky - 1984 r e Cockett, Cruttwell - 2014

Main References R. Blute, R. Cockett, R.A.G. Seely, Differential Categories , Mathematical Structures in Computer Science Vol. 1616, pp 1049-1083, 2006. R. Blute, R. Cockett, R.A.G. Seely, Cartesian Differential Categories , Theory and Applications of Categories 23, pp. 622-672, 2009

Where do Differential Categories come from? A short history from the horse’s mouth: “More recently, in a series of papers: T. Ehrhard, L. Regnier The differential λ -calculus. Theoretical Computer Science 309(1-3) (2003) 1-41. T. Ehrhard, L. Regnier Differential interaction nets Workshop on Logic, Language, Information and Computation (WoLLIC), invited paper. Electronic Notes in Theoretical Computer Science, vol. 123, March 2005, Elsevier. Ehrhard and Regnier introduced the differential λ -calculus and differential proof nets. Their work began with Ehrhard’s construction of models of linear logic in the category of K¨ othe spaces and finiteness spaces. They notes that these models had a natural notion of differential operator and made the key observation that the logic notion of linear (using arguments exactly once) coincided with the mathematical notion of linear transformation (which is essential to the notion of derivative, as the best linear approximation of a function). This observation is central to the decision to situate a categorical semantics for differential in appropriately endowed monoidal categories. Our aim in this paper is to provide a categorical reconstruction of the Ehrhard-Regnier differential structure. In order to achieve this we introduce the notion of differential category which captures the key structural components required for a basic theory of differentiation. ”

Differential Categories vs. Codifferential Categories To start we will go over the definition of the dual notion of a differential category: a codifferential category . Why? The intuition for codifferential categories is more closely related to commutative algebra.

Codifferential Categories - Definition A codifferential category (dual of a differential category) is: An additive symmetric monoidal category, (i) (ii) With an algebra modality, Equipped with a deriving transformation. (iii)

Codifferential Categories - Definition A codifferential category (dual of a differential category) is: An additive symmetric monoidal category , (i) (ii) With an algebra modality, Equipped with a deriving transformation. (iii)

Additive Symmetric Monoidal Categories - Definition Additive symmetric monoidal categories are symmetric commutative monoid enriched monoidal categories. 1 Composition is written diagrammatically, so fg is the map that does f then g

Additive Symmetric Monoidal Categories - Definition Additive symmetric monoidal categories are symmetric commutative monoid enriched monoidal categories. An additive category is a category X such that each hom-set X ( A , B ) is a commutative monoid with binary operation + and zero 0, that is, we can add parallel maps f + g and there is a zero map 0, and such that composition 1 preserves the additive structure: f ( g + h ) k = fgk + fhk f 0 = 0 = 0 f An additive symmetric monoidal category is a symmetric monoidal category which is also an additive category, such that the tensor product ⊗ preserves the additive structure: f ⊗ ( g + h ) = f ⊗ g + f ⊗ h ( f + g ) ⊗ h = f ⊗ h + g ⊗ h f ⊗ 0 = 0 0 ⊗ f = 0 Note that this definition does not assume biproducts or negatives. 1 Composition is written diagrammatically, so fg is the map that does f then g

Additive Symmetric Monoidal Categories - Examples Example Let REL be the category of sets and relations. Objects are sets X , and maps R : X → Y are subsets R ⊆ X × Y . REL is an additive symmetric monoidal category wher The monoidal structure is given by the Cartesian product of sets. Unit: {∗} Tensor product of objects X ⊗ Y := X × Y Tensor product of relations R ⊂ X × Y and S ⊆ A × B is R ⊗ S := { (( x , a ) , ( y , b )) | ( x , y ) ∈ R , ( y , b ) ∈ S } ⊆ ( X × A ) × ( Y × B ). The sum of maps R , S ⊆ X × Y is their union R + S := R ∪ S ⊆ X × Y . The zero maps are the empty subsets 0 := ∅ ⊆ X × Y . Example Let K be a field and and let VEC K to be the category of all K -vector spaces and K -linear maps between them. VEC K is additive symmetric monoidal category where: The monoidal structure is given by the tensor product of vector spaces ⊗ and the unit is K . The sum of K -linear maps f , g : V → W is the standard pointwise sum of linear maps, ( f + g )( v ) := f ( v ) + g ( v ). The zero maps 0 : V → W are the K -linear maps which map everything to zero. The additive structure in both examples are induced from finite biproducts.

Codifferential Categories - Definition A codifferential category (dual of a differential category) is: An additive symmetric monoidal category, (i) (ii) With an algebra modality , Equipped with a deriving transformation. (iii)

� � � � � � � � � � � Algebra Modality - Definition An algebra modality on a (strict) symmetric monoidal category is a monad (S , µ, η ) µ η � S A � S A SS A A equipped with two natural transformations m and u (where K is the unit): m u � S A � S A S A ⊗ S A K such that for every object A , (S A , m , u) is a commutative monoid (where σ is the symmetry): m ⊗ 1 � u ⊗ 1 σ � S A ⊗ S A S A S A ⊗ S A S A ⊗ S A ⊗ S A S A ⊗ S A S A ⊗ S A 1 ⊗ u m 1 ⊗ m m m m � S A � S A S A ⊗ S A S A ⊗ S A S A m m and µ is a monoid morphism: µ ⊗ µ u � S A ⊗ S A � SS A SS A ⊗ SS A K µ m m u � S A SS A S A µ

Algebra Modality - Example I Example Define the algebra modality M on REL as follows: for a set X , let M X be the free commutative monoid over a set X , equivalently the free N -module over X , or equivalently the set of finite multisets of X .

Algebra Modality - Example I Example Define the algebra modality M on REL as follows: for a set X , let M X be the free commutative monoid over a set X , equivalently the free N -module over X , or equivalently the set of finite multisets of X . Explicitly, for a function f : X → N define supp( f ) := { x ∈ X | f ( x ) � = 0 } . Then define M X as: M X = { f : X → N | | supp( f ) | < ∞} The monoid structure on M X is defined by point-wise addition, ( f + g )( x ) = f ( x ) + g ( x ), while the unit is u : X → N which maps everything to zero, u( x ) = 0. For each x ∈ X , let η x : X → N which maps y to 1 if x = y and 0 otherwise.

Algebra Modality - Example I Example Define the algebra modality M on REL as follows: for a set X , let M X be the free commutative monoid over a set X , equivalently the free N -module over X , or equivalently the set of finite multisets of X . Explicitly, for a function f : X → N define supp( f ) := { x ∈ X | f ( x ) � = 0 } . Then define M X as: M X = { f : X → N | | supp( f ) | < ∞} The monoid structure on M X is defined by point-wise addition, ( f + g )( x ) = f ( x ) + g ( x ), while the unit is u : X → N which maps everything to zero, u( x ) = 0. For each x ∈ X , let η x : X → N which maps y to 1 if x = y and 0 otherwise. Define the following natural transformations:     � η = { ( x , η x ) | x ∈ X } ⊆ X × M X µ =  ( F , f ) | F ∈ MM X  ⊆ MM X × M X f ∈ supp( F ) u = { ( ∗ , u) } ⊆ {∗} × M X m = { (( f , g ) , f + g ) | f , g ∈ M X } ⊆ (M X × M X ) × M X