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Can you Differentiate a Polynomial? Introduction Can you Differentiate a Polynomial? J.R.B. Cockett Department of Computer Science University of Calgary Alberta, Canada robin@cpsc.ucalgary.ca Halifax, June 2012 Can you Differentiate a

  1. Can you Differentiate a Polynomial? Introduction Can you Differentiate a Polynomial? J.R.B. Cockett Department of Computer Science University of Calgary Alberta, Canada robin@cpsc.ucalgary.ca Halifax, June 2012

  2. Can you Differentiate a Polynomial? Introduction WHAT IS THIS TALK ABOUT? PART I: Differential Categories PART II: Structural polynomials Part II concerns one of the motivating example for the development of Cartesian Differential Categories! ... of course, examples can be very confusing.

  3. Can you Differentiate a Polynomial? Differential categories The story so far .. ⊗ -differential categories ... ◮ ⊗ -Differential categories = Seely category + differential operator ◮ Simple categorical axiomatization ◮ Abstract framework for (additively enriched) differentiation ◮ Lots of sophisticated models ◮ Inspired by Ehrhard’s work: K¨ othe spaces, finiteness spaces and (with Regnier) on the differential λ -calculus ◮ The “linear algebra” approach to calculus. (developed with Blute and Seely)

  4. Can you Differentiate a Polynomial? Differential categories The story so far .. × -differential categories ... ⊗ -differential categories are not enough! The following are in a Cartesian rather than linear world: ◮ Classical multivariable differential calculus ... ◮ Differential lambda calculus (Ehrhard, Renier, ... – French School) ◮ Combinatoric species differentiation (Joyal, Bergeron,.. – Montreal School) ◮ Differentiation of data types (McBride, Gahni, Fiori, .. – UK School) ◮ CoKleisli category of a ⊗ -differential category ... COMING SHORTLY ON A BIG SCREEN NEAR YOU: THEIR CATEGORICAL AXIOMATIZATION!!

  5. Can you Differentiate a Polynomial? Differential categories The story so far .. Many more differential categories ... Aside: Cartesian differential categories are not enough either ... ◮ Classical differential calculus considers partial maps ... ◮ Calculus on manifolds uses topological notions .... ◮ Manifolds and varieties in algebraic geometry ... ◮ Synthetic differential geometry .. HOW ARE THESE AXIOMATIZED?

  6. Can you Differentiate a Polynomial? Differential categories The story so far .. Many more differential categories ... ◮ Differential restriction categories (Cockett, Crutwell, Gallagher) ◮ Categories with tangent structure (Rosicky; Cockett, Crutwell) Tangent structure is the most abstract of them all ... .... and includes them all. In particular, they provide link to synthetic differential calculus .... One ring to rule them all ...

  7. � ��� Can you Differentiate a Polynomial? Differential categories The story so far .. The story so far ... Tangent Rest Cats � � � � � � � � � � � � � � � � � � Tangent Categories Differential Rest Cats � � � � � � � � � � � � � � � � � � � Cartesian Diff CAts coKleisli Monoidal Diff Cats

  8. Can you Differentiate a Polynomial? Differential categories Arriving at axioms Axioms for a cartesian differential ... CARTESIAN DIFFERENTIAL CATEGORIES ARE ESSENTIALLY THE coKLEISLI CATEGORIES OF ⊗ -DIFFERENTIAL CATEGORIES. ... just write down the equations ... HOW HARD CAN THAT BE? ... generated two papers (and counting) so far!

  9. Can you Differentiate a Polynomial? Differential categories Arriving at axioms It took a long time to get it right! WHY? A. We were idiots? B. Academic baggage ... C. Calculus for the masses ... D. The structure of the area has been trampled on with: ◮ Preconceptions: infinitesimals and dx .... ◮ Manipulations without algebraic basis ... ◮ Notational short-cuts which mask structure ... E. The axioms are actually a little tricky!

  10. Can you Differentiate a Polynomial? Differential categories Arriving at axioms DID WE GET IT RIGHT?

  11. Can you Differentiate a Polynomial? Differential categories Arriving at axioms The good news: We are confident we have basic axiomatization right! FINALLY! ... and people are beginning to use it!

  12. Can you Differentiate a Polynomial? Differential categories Arriving at axioms The bad news: How do we know? ◮ capture coKleisli categories of diff cats ◮ captures key examples ◮ Fa` a di Bruno construction gives a comonadic description ... NOT EVERYONE IS CONVINCED examples are very confusing! .... effort needed to avoid reinventing the wheel!!

  13. Can you Differentiate a Polynomial? Differential categories Arriving at axioms CARTESIAN DIFFERENTIAL CATEGORIES To formulate cartesian differential categories need: (a) Left additive categories (b) Cartesian structure in the presence of left additive structure (c) Differential structure Example to have in mind: vector spaces with smooth functions ....

  14. Can you Differentiate a Polynomial? Differential categories Left-additive categories Left-additive categories A category X is a left-additive category in case: ◮ Each hom-set is a commutative monoid (0,+) ◮ f ( g + h ) = ( fg ) + ( fh ) and f 0 = 0 each f is left additive .. g f − − → A − − → B C − − → h A map h is said to be additive if it also preserves the additive structure on the right ( f + g ) h = ( fh ) + ( gh ) and 0 h = 0. f − − → h A B − − → C − − → g NOTE: additive maps are the exception ...

  15. Can you Differentiate a Polynomial? Differential categories Left-additive categories Lemma In any left additive category: (i) 0 maps are additive; (ii) additive maps are closed under addition; (iii) additive maps are closed under composition; (iv) identity maps are additive; (v) if g is a retraction which is additive and the composite gh is additive then h is additive; (vi) if f is an isomorphism which is additive then f − 1 is additive. Additive maps form a subcategory ...

  16. Can you Differentiate a Polynomial? Differential categories Left-additive categories Example (i) The category whose objects are commutative monoids CMon but whose maps need not preserve the additive structure. (ii) Real vector spaces with smooth maps. (iii) The coKleisli category for a comonad on an additive category when the functor is not additive.

  17. Can you Differentiate a Polynomial? Differential categories Left-additive categories Products in left additive categories A Cartesian left-additive category is a left-additive category with products such that: ◮ the maps π 0 , π 1 , and ∆ are additive; ◮ f and g additive implies f × g additive. All our earlier example are Cartesian left-additive categories!

  18. Can you Differentiate a Polynomial? Differential categories Left-additive categories Lemma The following are equivalent: (i) A Cartesian left-additive category; (ii) A left-additive category for which X + has biproducts and the the inclusion I : X + − → X creates products; (iii) A Cartesian category X in which each object is equipped with a chosen commutative monoid structure (+ A : A × A − → A , 0 A : 1 − → A ) such that + A × B = � ( π 0 × π 0 )+ A , ( π 1 × π 1 )+ B � and 0 A × B = � 0 A , 0 B � .

  19. Can you Differentiate a Polynomial? Differential categories Left-additive categories Lemma In a Cartesian left-additive category: (i) f is additive iff ( π 0 + π 1 ) f = π 0 f + π 1 f : A × A − → B and 0 f = 0 : 1 − → B ; (ii) g : A × X − → B is additive in its second argument iff 1 × ( π 0 + π 1 ) g = (1 × π 0 ) g + (1 × π 1 ) g : A × X × X − → B (1 × 0) g = 0 : A × 1 − → B . “Multi-additive maps” are maps additive in each argument.

  20. Can you Differentiate a Polynomial? Differential categories Left-additive categories A functor between Cartesian left-additive categories is left-additive in case F ( f + g ) = F ( f ) + F ( g ) and F (0) = 0. Lemma A left-additive functor, F : X − → Y , necessarily preserves products F ( A × B ) ≡ F ( A ) × F ( B ) , additive maps and multi-additive maps. The category of all cartesian left-additive categories and left-additive functors is CLAdd.

  21. Can you Differentiate a Polynomial? Differential categories Differential Structure An operator D × on the maps of a Cartesian left-additive category f X − − → Y X × X − − − − → Y D × [ f ] is a Cartesian differential operator in case it satisfies: [CD.1] D × [ f + g ] = D × [ f ] + D × [ g ] and D × [0] = 0; [CD.2] � ( h + k ) , v � D × [ f ] = � h , v � D × [ f ] + � k , v � D × [ f ]; [CD.3] D × [1] = π 0 , D × [ π 0 ] = π 0 π 0 , and D × [ π 1 ] = π 0 π 1 ; [CD.4] D × [ � f , g � ] = � D × [ f ] , D × [ g ] � (and D × [ �� ] = �� ); [CD.5] D × [ fg ] = � D × [ f ] , π 1 f � D × [ g ]. [CD.6] �� f , 0 � , � h , g �� D × [ D × [ f ]] = � f , h � D × [ f ]; [CD.7] �� 0 , f � , � g , h �� D × [ D × [ f ]] = �� 0 , g � , � f , h �� D × [ D × [ f ]] A Cartesian left-additive category with such a differential operator is a Cartesian differential category .

  22. Can you Differentiate a Polynomial? Differential categories Differential Structure What was so hard about that? ANSWER: the last two rules!! ◮ They are independent ... ◮ They involve higher differentials ... ◮ Not so obvious where they come from ...

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