preamble Differential Join Restriction Categories Jonathan - PowerPoint PPT Presentation

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Differential Join Restriction Categories Jonathan Gallagher with Robin Cockett and Geoff Cruttwell October 23, 2010 1 / 24

Talk Outline • Talk Outline Goal: Give and motivate the definition of differential Background join restriction category. Differential Restriction Categories Differential Join Restriction Categories Here are the ideas outlining the talk. • Restriction categories axiomatize partiality. • Cartesian differential categories axiomatize smooth functions on R n . • Differential restriction categories combine the two theories to axiomatize the category of smooth functions on an open subset of R n . • Differential join restriction categories bring more topological structure. 2 / 24

• Talk Outline Background • Restriction Categories • Restriction Categories Examples • Cartesian Differential Categories • Differential Restriction Categories • Cartesian Restriction Background Categories • Cartesian Left Additive Restriction Categories Differential Restriction Categories Differential Join Restriction Categories 3 / 24

Restriction Categories • Talk Outline Definition 1. A restriction category is a category X with a Background combinator, ( ) : X ( A, B ) − → X ( A, A ) , satisfying • Restriction B A Categories R.1 f f = f ; f • Restriction Categories Examples • Cartesian Differential R.2 f g = g f ; Categories • Differential Restriction Categories R.3 f g = f g ; • Cartesian Restriction A Categories • Cartesian Left f Additive Restriction R.4 fh = fh f . Categories Differential Restriction Categories Differential Join Restriction Categories 4 / 24

Restriction Categories Examples • Talk Outline • P AR the category of sets and partial functions is a restriction Background category. f gives the domain of definition of f . • Restriction Categories • Restriction � Categories Examples f ( x ) ↓ x • Cartesian Differential f ( x ) = Categories ↑ else • Differential Restriction Categories • Cartesian Restriction Categories • Cartesian Left • T OP the category of topological spaces and continous maps Additive Restriction Categories defined on an open set is a restriction category. This category Differential Restriction Categories has the same restriction as P AR . Differential Join Restriction Categories Other examples of restriction categories can be found in [2] 5 / 24

Cartesian Differential Categories Cartesian Differential Categories [1] axiomatize smooth functions on R n by • Talk Outline Background axiomatizing a differential combinator (think Jacobian). The differential • Restriction combinator has the type, Categories • Restriction Categories Examples • Cartesian Differential f : R n − → R m Categories • Differential Restriction D [ f ] : R n − → ( R n ⊸ R m ) Categories • Cartesian Restriction Categories It is too strong to assume that the category is closed with respect to linear • Cartesian Left Additive Restriction maps. Thus the differential combinator is used in uncurried form. Categories Differential Restriction Categories f : R n − → R m Differential Join D [ f ] : R n × R n − Restriction Categories → R m The first coordinate is the directional vector. The second coordinate is the point of differentiation. This axiomatization will require products. Left additivity is needed for vectors. 6 / 24

Differential Restriction Categories • Talk Outline Background • Restriction Categories To build the theory of differential restriction categories, change the • Restriction Categories Examples theory of cartesian differential categories in light of restriction • Cartesian Differential Categories structure. This means reconsidering: • Differential Restriction Categories • Cartesian Restriction Categories • Cartesian Left • Cartesian categories, Additive Restriction Categories Differential Restriction • Left additive categories and cartesian left categories, and Categories Differential Join • Differential categories. Restriction Categories 7 / 24

� � � Cartesian Restriction Categories • Talk Outline Pairing two maps together in a restriction category brings up the partiality of both. Background • Restriction Categories Definition 2. A map in a restriction category is total when f = 1 . • Restriction Categories Examples • Cartesian Differential Definition 3. A restriction product of A, B is an object A × B such that Categories • Differential Restriction for any f : C − → A, g : C − → B there is a unique � f, g � : C − → A × B Categories • Cartesian Restriction such that Categories • Cartesian Left a ≤ b ⇔ a b = a Additive Restriction C Categories � � � ��������� � f g � Differential Restriction � � f,g � � � Categories ≥ ≤ � � � B Differential Join A × B A Restriction Categories π 0 π 1 where π 0 , π 1 are total and � f, g � = f g . A restriction terminal object is 1 such that for any object A , there is a unique total map ! A : A − → 1 which satisfies ! 1 = id 1 . Further, for any map f : A − → B , f ! B ≤ ! A . A cartesian restriction category has all restriction products. 8 / 24

Cartesian Left Additive Restriction Categories • Talk Outline The addition of two maps must only be defined when both are. Background • Restriction Definition 4. A left additive restriction category has each Categories • Restriction X ( A, B ) a commutative monoid with f + g = f g and 0 being Categories Examples total. Furthermore, h ( f + g ) = hf + hg and s 0 = s 0 • Cartesian Differential Categories • Differential Restriction Categories Definition 5. A map, h , in a left additive restriction category is total • Cartesian Restriction additive if h is total, and ( f + g ) h = fh + gh . Categories • Cartesian Left Additive Restriction Categories Definition 6. A cartesian left additive restriction category is both Differential Restriction a left additive restriction category and a cartesian restriction Categories category where π 0 , π 1 , and ∆ are total additive, and Differential Join Restriction Categories ( f + h ) × ( g + k ) = ( f × g ) + ( h × k ) . 9 / 24

• Talk Outline Background Differential Restriction Categories • Differential Restriction Categories • Differential Restriction Categories • Differential Restriction Categories: Examples Differential Restriction • Differential Restriction Categories: Examples Categories • Differential Restriction Categories: Examples Differential Join Restriction Categories 10 / 24

Differential Restriction Categories • Talk Outline A differential restriction category is a cartesian left additive Background restriction category with a differential combinator Differential Restriction Categories • Differential Restriction Categories f : X − → Y • Differential Restriction Categories D [ f ] : X × X − → Y • Differential Restriction Categories: Examples • Differential Restriction such that Categories: Examples • Differential Restriction DR.1 D [ f + g ] = D [ f ] + D [ g ] and D [0] = 0 (additivity of the Categories: Examples differential combinator) ; Differential Join Restriction Categories DR.2 � g + h, k � D [ f ] = � g, k � D [ f ] + � h, k � D [ f ] and � 0 , g � D [ f ] = gf 0 (additivity of differential in first coordinate) ; DR.3 D [1] = π 0 , D [ π 0 ] = π 0 π 0 , and D [ π 1 ] = π 0 π 1 ; DR.4 D [ � f, g � ] = � D [ f ] , D [ g ] � ; DR.5 D [ fg ] = � D [ f ] , π 1 f � D [ g ] (Chain rule) ; 11 / 24

Differential Restriction Categories • Talk Outline f : X − → Y Background D [ f ] : X × X − → Y Differential Restriction Categories • Differential Restriction (... and) Categories • Differential Restriction Categories DR.6 �� g, 0 � , � h, k �� D [ D [ f ]] = h � g, k � D [ f ] (linearity of the • Differential Restriction derivative) Categories: Examples • Differential Restriction Categories: Examples DR.7 �� 0 , h � , � g, k �� D [ D [ f ]] = �� 0 , g � , � h, k �� D [ D [ f ]] (independence • Differential Restriction Categories: Examples of partial derivatives) ; Differential Join DR.8 D [ f ] = (1 × f ) π 0 ; Restriction Categories DR.9 D [ f ] = 1 × f (Undefinedness comes from the “point”) . 12 / 24

Differential Restriction Categories: Examples Example 1: Smooth Maps on open subsets of R n • Talk Outline Background • The Jacobian matrix provides the differential structure. Differential Restriction Categories • Differential Restriction ∂f 1 ∂f 1 Categories   ∂x 1 ( y 1 , . . . , y n ) . . . ∂x n ( y 1 , . . . , y n ) • Differential Restriction . . Categories ... J f ( y 1 , . . . , y n ) = . .   • Differential Restriction . .   Categories: Examples ∂f m ∂f m ∂x 1 ( y 1 , . . . , y n ) . . . ∂x n ( y 1 , . . . , y n ) • Differential Restriction Categories: Examples • Differential Restriction Categories: Examples Differential Join • D [ f ] : ( x 1 , . . . , x n , y 1 , . . . , y n ) �→ Restriction Categories J f ( y 1 , . . . , y n ) · ( x 1 , . . . , x n ) . 13 / 24