CARTESIAN DIFFERENTIAL CATEGORIES

Transcription

1 Theory and Applications o Categories, Vol. 22, No. 23, 2009, pp CARTESIAN DIFFERENTIAL CATEGORIES R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY Abstract. This paper revisits the authors notion o a dierential category rom a dierent perspective. A dierential category is an additive symmetric monoidal category with a comonad (a coalgebra modality ) and a dierential combinator. The morphisms o a dierential category should be thought o as the linear maps; the dierentiable or smooth maps would then be morphisms o the cokleisli category. The purpose o the present paper is to directly axiomatize dierentiable maps and thus to move the emphasis rom the linear notion to structures resembling the cokleisli category. The result is a setting with a more evident and intuitive relationship to the amiliar notion o calculus on smooth maps. Indeed a primary example is the category whose objects are Euclidean spaces and whose morphisms are smooth maps. A Cartesian dierential category is a Cartesian let additive category which possesses a Cartesian dierential operator. The dierential operator itsel must satisy a number o equations, which guarantee, in particular, that the dierential o any map is linear in a suitable sense. We present an analysis o the basic properties o Cartesian dierential categories. We show that under modest and natural assumptions, the cokleisli category o a dierential category is Cartesian dierential. Finally we present a (sound and complete) term calculus or these categories which allows their structure to be analysed using essentially the same language one might use or traditional multi-variable calculus. 0. Introduction Over the past ew centuries, one o the most undamental concepts in all o mathematics has been dierentiation. In recent decades several attempts have been made to abstract this notion, including approaches based on geometric, algebraic, and logical intuitions. The approach o the current paper is categorical, insoar as we wish to consider axiomatizations o categories which have suicient structure to deine dierentiation o maps. Any additional categorical structure, e.g. monoidal, Cartesian, or comonadic, exists in support o dierentiation. In [BCS 06], the current authors introduced the notion o a dierential category to provide a basic axiomatization or dierential operators in monoidal categories. The initial impetus or the deinition came rom work o Ehrhard and Regnier [Ehrhard & Regnier 05, Ehrhard & Regnier 03], who deined irst a notion o dierential λ-calculus Received by the editors and, in revised orm, Transmitted by Robert Paré. Published on Mathematics Subject Classiication: 18D10,18C20,12H05,32W99. Key words and phrases: monoidal categories, dierential categories, Kleisli categories, dierential operators. c R.F. Blute, J.R.B. Cockett and R.A.G. Seely, Permission to copy or private use granted. 622

2 CARTESIAN DIFFERENTIAL CATEGORIES 623 and subsequently dierential proo nets. The dierential λ-calculus is an extension o simply-typed λ-calculus with an additional operation allowing the dierentiation o terms. Dierential proo nets are a (graph-theoretic) syntax or linear logic extended with a dierential operator on proos. An important eature o their systems, not precluded in ours, is that in one setting, they combine the essence o both calculus and computability. Their work grew out o the development o models o linear logic (examples o - autonomous categories) in which there was a natural dierential operator, such as Köthe spaces [Ehrhard 01]. Every model o linear logic comes equipped with a monad, the storage operator, rom which the cokleisli category arises. One can then abstract away rom models o linear logic, retaining the comonad as the key eature. From this perspective it is a very natural step to consider the cokleisli categories o such models, not least because it is this setting which supports a calculus which much more closely resembles the elementary dierential calculus with which we are all amiliar. The calculus one obtains rom this perspective is quite dierent than the dierential lambda-calculus o Ehrhard and Regnier, in that it is directly inspired by the cokleisli structure. And so this let open the question o how to characterize this situation. The notion o a dierential category provides a basic axiomatization or dierential operators in monoidal categories, which not only generalizes the work o Ehrhard and Regnier but also captures the standard elementary models o dierential calculus and provides a theoretical substrate or studying a number o less standard examples. The structure necessary to support dierentiation in [BCS 06] is an additive, monoidal category with a coalgebra modality. (These terms will be reviewed below.) The morphisms in a dierential category should be thought o as linear maps, with maps in the cokleisli category being the smooth maps. Then the dierential operator is o the ollowing type: : SA B D []: A SA One then writes down suitable axioms which such an operator must satisy. In particular, we have the Leibniz rule, the chain rule, and other basic rules o dierentiation expressed coalgebraically. The goal o the present paper is to develop an axiomatisation which directly characterizes the smooth maps: in other words, to characterize the cokleisli structure o dierential categories directly. This leads us to the notion o a Cartesian dierential category. This notion embodies the multi-variable dierential calculus which, being a undamental tool o modern mathematics, is well worth studying in its own right. The basic structure needed or Cartesian dierential categories is simpler than is needed or dierential categories: just a let additive category with inite products. The dierential operator takes on the ollowing orm X Y X X D [] However the necessary equations turn out to be more complicated, and the passage rom the cokleisli category o a dierential category to a Cartesian dierential category Y B

3 624 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY turns out to be surprisingly subtle. While we do describe the conditions under which a dierential category gives rise to a Cartesian dierential category as its cokleisli category, a ull characterization o this situation is let to a sequel. The main result in this direction is Proposition The organization o the paper is as ollows. Fundamental to dierentiation is the ability to add maps: however, the settings in which we are interested are not additively enriched. Thereore the irst section is dedicated developing the general theory o let additive categories. In the second section we introduce the key notion o a Cartesian dierential operator and the equations it satisies. In this section, we also show how most o the axiomatization o these categories is determined by requiring that their bundle categories behave in the expected manner. These are ibrations which carry the dierential structure, insoar as the composition o maps is just the chain rule. It is possible to develop higher-order bundle categories in which composition is determined by the higherorder (Faà di Bruno) chain rules. These higher-dierential bundle categories completely determine all the axiomatization o Cartesian dierential categories, but developing this would require more technical apparatus than seems justiied in this paper, so will be presented elsewhere. As always, we are interested in graphical representations o morphisms in ree categories, an approach begun in [BCST 97]. But in this paper we ocus more on a term calculus. This calculus can be seen as eectively reimagining the traditional dierential calculus as a rewrite system. Section 4 o this paper is devoted to analyzing this system in detail. In particular, we show the system s soundness and completeness. The term calculus allows or an elegant description o ree Cartesian dierential categories, which we shall present in a sequel. The reader should keep one key simple example in mind when reading the paper: the category o smooth maps, which behaves just as one would expect rom a irst year calculus course. Objects are natural numbers and maps n m are smooth maps IR n IR m. It is easiest to describe the dierential operator via an example. Suppose we have a smooth map : 3 1, such as (x, y, z) = xyz. The Jacobian o this map is [yz, xz, xy] which may be regarded as a smooth operator which assigns a linear operator to any point (x, y, z). This is how we get a smooth map D (): IR 3 IR 3 IR, which is linear in the irst variable (in the irst triple o coordinates): D (): ((u, v, w), (r, s, t)) (st, rt, rs) (u, v, w) = stu + rtv + rsw In act, the notation we will introduce in our term calculus is slightly dierent than this, and keeps more careul track o ree vs. bound variables. This leads to one o our axioms or Cartesian dierential categories: the unction D[] is linear in its irst n variables. The rest o the axioms express other aspects o dierentiability. The oundations or dierential calculus may be approached in at least three undamentally dierent ways. At one extreme is the synthetic approach where one wishes to create a calculus so deeply embedded into the underlying set theory that one is willing to

4 CARTESIAN DIFFERENTIAL CATEGORIES 625 turn the world o mathematics and philosophy upside down to ensure that everything becomes dierentiable. Somewhere nearer the middle is the Platonic approach which takes the view that the ability to dierentiate ultimately devolves onto the topological and limiting properties o a single crucial object: the real line. And then there is the mechanical approach. Devoid o overarching philosophy or greater purpose, it takes calculus as a system which, like any other, should immediately be taken apart to determine how the behavior o one part depends on structure in some other part. Dismembering calculus in this manner, it seeks to reuse these structures in outlandish conigurations elsewhere. This work belongs unapologetically to this last camp. It seeks to isolate some basic structural properties which give rise to behaviors reminiscent o the dierential calculus. We believe abstracting dierential calculus in this manner serves a useul purpose which also relects developments in other areas. Dierentials (over arbitrary base rings) are used non-trivially in various areas o algebraic geometry, and dierentials also appear in dierent guises in both combinatorics and computer science. A ramework uniying such notions is useul, and in particular allows one to distinguish clearly between what is generic and what is speciic. 1. Let additive categories The purpose o this section is to develop the basic theory o let additive categories which underlies the theory o Cartesian dierential categories. As a basic example consider the category o commutative monoids with morphisms which are arbitrary maps which ignore the additive structure. Despite the act that additive structure is being ignored, the maps between any two objects have a natural additive structure, given by pointwise addition ( +g)(x) = (x)+g(x). Furthermore, this additive structure is preserved by composition on the let h( + g) = h + hg (throughout the paper we use the diagrammatic order o composition, sometimes denoted with a semicolon, oten just by juxtaposition). A category with the property that each homset is a commutative monoid and or which composition on the let preserves this structure is called a let additive category. This category may be viewed rather dierently: it is the cokleisli category with respect to the comonad generated rom the comonad on commutative monoids generated by the composite o the underlying unctor and the ree unctor. This illustrates an important way in which let additive structure arises: any (non-additive) comonad on an additive (or indeed let-additive) category always has its cokleisli category let-additive. Another basic example o a let additive category which is central to this paper consists o the category whose objects are inite dimensional real vector spaces and whose maps are (ininitely) dierentiable maps. These maps allow pointwise addition and so certainly orm a let additive category. In addition, this category has a dierential structure which is the main subject matter o the paper and is discussed in the next section The basic deinition.

5 626 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY Deinition. A category X is let additive 1 in case each hom-set is a commutative monoid and (g +h) = (g)+(h) and 0 = 0. A map h in a let additive category is said to be additive i it also preserves the additive structure o the hom-set on the right ( + g)h = (h) + (gh) and 0h = 0. In general additive maps will be the exception in a let additive category. However, the additive maps orm an interesting subcategory: Proposition. In any let additive category, 1. 0 maps are additive; 2. additive maps are closed under addition; 3. additive maps are closed under composition; 4. all identity maps are additive; 5. i m is an additive monic and m is additive then is additive; 6. i g is a retraction which is additive and the composite gh is additive then h is additive; 7. i r is a retraction with section m so that the idempotent rm is additive, then r is additive i its section m is additive; 8. i is an isomorphism which is additive, then 1 is additive. Proo. (1): ( + g)0 = 0 = = 0 + g0. (2): I and g are additive then (x+y)( +g) = (x+y) +(x+y)g = x +y +xg+yg = x( + g) + y( + g). (3): I and g are additive then (x + y)g = (x + y)g = xg + yg. (4,5): Immediate. (6): This is slightly more subtle: suppose g is additive and a retraction, so that there is a g with g g = 1, and gh is additive then, as (x + y) = (xg g + yg g) = (xg + yg )g then (x + y)h = (xg + yg )gh = xg gh + yg gh = xh + yh. (7): Since rm is additive, by (5) i m is additive, then r is additive; and by (6) i r is additive then m is additive. (8): I is an isomorphism it is certainly a retraction and 1 = 1 is certainly additive so by the previous property 1 is additive. 1 We should emphasize that our additive categories are commutative monoid enriched categories, rather than Abelian group enriched; some people might preer to call them semi-additive. Furthermore, we do not require biproducts as part o the structure at this stage. In particular, our deinition is not the same as the one in [[M 71]].

6 CARTESIAN DIFFERENTIAL CATEGORIES 627 Note that it is certainly not the case that all isomorphisms are additive. However, we can conclude: Corollary. The additive maps o a let additive category X orm an additive subcategory whose inclusion I: X + X relects isomorphisms Example. The category CMon o commutative monoids with set maps (i.e. without any preservation properties) is let additive, but generally its maps are not additive. Let additivity is easily shown: (g + h)(x) = (g + h)((x)) = g((x)) + h((x)) = (g)(x) + (h)(x) = (g + h)(x) (and 0 is similar). As an example o the ailure o additivity, however, consider that (0)(x) = (0(x)) = (0), and we have explicitly not required (0) = 0 (and similarly or addition). The additive maps in this setting are just the commutative monoid homomorphisms Cartesian let additive categories. Our main interest centres around let additive categories which have products which behave coherently with respect to the additive structure: Deinition. A Cartesian let additive category is a let additive category with products such that the structure maps π 0, π 1, and are additive and that whenever and g are additive then g is additive. Notice that i and g are additive then, g is additive as (x + y), g = (x + y)( g) = x( g) + y( g) = x, g + y, g Conversely one may replace the requirement that is additive and that preserves additivity by the single requirement that pairing preserves additivity as both can be expressed by pairing additive maps. Thus, equivalently, a category is Cartesian let additive in case it has products or which the projections are additive and whenever and g are additive then, g is additive. Cartesian let additive categories can be ormulated in various other ways as well: Proposition. The ollowing are equivalent: (i) A Cartesian let additive category; (ii) A let additive category with products such that all projections and pairings o additive maps are additive. (iii) A let additive category or which X + has biproducts and the the inclusion I: X + creates products; X (iv) A Cartesian category X in which each object is equipped with a chosen commutative monoid structure compatible with products: (+ A : A A A, 0 A : 1 A) satisying + A B = (π 0 π 0 )+ A, (π 1 π 1 )+ B and 0 A B = 0 A, 0 B.

7 628 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY Proo. (i) (ii) Above. (ii) (iii) Clearly as the product structure is additive the category o additive maps will have products (and so biproducts). Further, the inclusion unctor will clearly create products. (iii) (iv) Deine + A = π 0 + π 1 and 0 A = 0 then this gives each object an additive structure. Note that + A B = π 0 + π 1 = (π 0 + π 1 ) π 0, π 1 = π 0 π 0 + π 1 π 0, π 0 π 1 + π 1 π 1 = (π 0 π 0 )+ A, (π 1 π 1 )+ B (iv) (i) Deine + g =, g + B then this certainly makes the category let additive. Furthermore, each π i is additive as (, + g, g )π 0 =,, g, g (π 0 π 0 )+ =, π 0 + g, g π 0. Clearly maps are additive in case they are homomorphisms o the chosen additive structure: but this means i and g are additive then, g is additive as the additive structure on the product is the product additive structure! The last characterization does rely on the choice o product structure. The equivalence to being Cartesian let additive inorms us that the choice can be made up to additive isomorphism. For, seen as a let additive category, there is one global structure albeit it may be represented locally in a variety o coherent ways. The act that an additive map : A B in a Cartesian let additive category is precisely a homomorphism o the additive structure A A B B + A A B suggests the simple test or additivity in part (ii) o the ollowing technical lemma: + B

8 CARTESIAN DIFFERENTIAL CATEGORIES Lemma. In a Cartesian let additive category: (i), g +, g = +, g + g and 0 = 0, 0 ; (ii) is additive i and only i (π 0 + π 1 ) = π 0 + π 1 : A A B and 0 = 0: 1 B; (iii) g: A X B is additive in its second argument i and only i 1 (π 0 +π 1 )g = (1 π 0 )g+(1 π 1 )g: A X X B and (1 0)g = 0: A 1 B. Being additive in the second argument means x, y + z g = x, y g + x, z g and x, 0 g = 0; being additive in an argument is a property which will become more central shortly. Proo. (i) We have the ollowing calculation:, g +, g = (, g +, g ) π 0, π 1 = (, g +, g )π 0, (, g +, g )π 1 =, g π 0 +, g π 0,, g π 1 +, g π 1 = +, g + g and a similar calculation or the zero. (ii) I is additive this equality holds. Conversely (h + k) = h, k (π 0 + π 1 ) = h, k (π 0 + π 1 ) = h + k. (iii) Similar. One reason or demanding that the product structure be additive is the ollowing (which uses the act that products in additive categories are necessarily biproducts): Corollary. For any Cartesian let additive category X the subcategory o the additive maps I: X + X has biproducts. Again we may consider the example CMon: it is clear that letting the product be the usual product o commutative monoids will ensure that we obtain a Cartesian let additive category. Note that an arbitrary monoid which has base set the product o the base sets o M 1 and M 2 will not work as the let additive product as π 0 + π 1 will no longer give the monoid operation.

9 630 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY 1.3. Cartesian let additive unctors Deinition. A unctor between Cartesian let additive categories is Cartesian let additive in case F ( + g) = F () + F (g) and F (0) = 0; F preserves products (i.e F (A) F (π 0 ) F (A B) F (π 1 ) F (B) is a product). Clearly the identity unctor is let additive and we may compose Cartesian let additive unctors, thus, this, together with natural transormations (whose components are additive) gives the data or a 2-category. Notice that Cartesian let additive unctors pre- serve the additive structure maps A A + A 0 structure, we have: 1 so, crucially or the 2-dimensional Lemma. A let additive unctor is a Cartesian let additive unctor i and only i it preserves additive maps. Proo. A map is additive i and only i it is a homomorphism o the given additive structure: this property is preserved by Cartesian let additive unctors. The converse ollows as biproducts are equationally deined using the additive maps. Suppose S = (S, δ, ) is any comonad (where S need not be Cartesian let additive) on a Cartesian let additive category, X. Then clearly the cokleisli maps S(X) Y inherit an addition rom X and with this X S becomes let additive. (In the ollowing calculation, we shall distinguish maps in the cokleisli category by setting them in boldace; maps in X will be in ordinary type.) (g 1 + g 2 ) = δs()(g 1 + g 2 ) = δs()g 1 + δs()g 2 = g 1 + g 2 As X S is a cokleisli category it has products inherited rom X with X π 0 X Y π 1 Y = X π 0 S(X Y ) π 1 Y. Note that these projections are additive in X S as ( + g)π 0 = δs( + g)π 0 = ( + g)π 0 = (π 0 ) + (gπ 0 ) = (π 0 ) + (gπ 0 ) Consider the pairing map: Z,g X Y = S(Z) and suppose and g are additive in X S then,g X Y (x + y),g = δs(x + y), g = δs(x + y), δs(x + y)g = (x + y), (x + y)g = x + y, xg + yg = x, xg + y, yg = x, g + y, g We thereore have:

10 CARTESIAN DIFFERENTIAL CATEGORIES Proposition. I S = (S, δ, ) is any comonad on a (Cartesian) let additive category X then X S is (Cartesian) let additive. Furthermore the canonical right adjoint G S : X X S is a Cartesian let additive unctor. Proo. The hard work was done above! It remains to check that G S is additive as it clearly preserves products; or this we need G S ( + g) = ( + g) = + g = G S () + G S (g) and G S (0) = 0 = 0. As an application o Proposition 1.3.3, we note that since a Cartesian let additive category X has products, or each object A X the unctor A is a comonad (using the act that A is canonically a comonoid); the cokleisli category is sometimes known as the simple slice category at A ; we shall denote it X[A]: Deinition. X[A] (called the simple slice category at A ) has as objects the objects o X, and as morphisms : X Y morphisms : X A Y o X. Identities are given by projections, and composition is the cokleisli composition: X Y g Z = X A 1 X A A 1 Y A g Z Corollary. Each simple slice X[A] o a Cartesian let additive category X is a Cartesian let additive category. Notice that the additive unctions in X[A] are exactly the maps : X A Y which are additive in their irst argument in the sense that x + y, z = x, z + y, z and 0, z = Cartesian closed let additive categories. A let additive category is a Cartesian closed let additive category in case it is a Cartesian let additive category which is Cartesian closed, so A is a let adjoint with right adjoint A, such that the passage: A X Y X curry() A Y preserves the additive structure. That is curry( +g) = curry()+curry(g) and curry(0) = Lemma. A Cartesian let additive category X is a Cartesian closed let additive category i and only i X is Cartesian closed and + A B =π 0 +π 1 A B A B A B k A + B =A (π 0 +π 1 ) A (B B) k 1 1 A 1 0 A 0 A B commute.

11 632 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY Proo. Only i is obvious by considering adjoints. For the converse, we note that when the stated condition holds we have: curry( + g) = curry(, g )(A + B ) = curry(), curry(g) k (A + B ) = curry(), curry(g) + A B = curry() + curry(g) so that addition is preserved by currying: the preservation o zero is similar. Thus, the category is Cartesian closed let additive. When a category is Cartesian closed then (A, η, µ) is a monad or each object A where A X π 1 X X A Y η curry(π1 ) 1 A (A (A X)) 1 eval A A (A (A X)) A (A X) A (A X) A X curry(( 1)(1 eval)eval) eval X µ The Kleisli category or this monad is clearly isomorphic to X[A] and the coherence requirement or closedness ensures that the two additive structures inherited rom X, regarding it as a Kleisli and cokleisli category respectively, coincide. In particular note that in a Cartesian closed let additive category A X Y is additive in its second argument i and only i X curry() A Y is additive. Once again we may consider CMon: this is a Cartesian closed let additive category provided one endows the hom-sets with the pointwise additive structure + g = λx.(x) + g(x) Additive bundle ibrations. O course, there is a well-known ibration associated with the simple slice categories; we are interested in the subibration whose ibres are just the additive parts o the simple ibres. We think o this ibration as the ibration o additive bundles over X, and so denote it by p: ABun(X) X. The objects o ABun(X) are pairs (X, A), its morphisms (X, A) (Y, B) are pairs (F, ) o morphisms o X, where F : X A Y is additive in its irst argument, and : A B. Composition is deined by (F, )(G, g) = ( F, π 1 G, g), and the identities are (π 0, 1 A ). We shall check (below) that i F, G are additive in the irst variable, so is F, π 1 G. ABun(X) has additive structure, deined coordinate-wise: (F, )+(G, g) = (F +G, + g), 0 = (0, 0). It also has products: 1 = (1, 1) and (X, A) (π 0 π 0,π 0 ) (X Y, A B) (π 0 π 1,π 1 ) (Y, B) In act: Proposition. I X is Cartesian let additive, then ABun(X) is as well.

12 CARTESIAN DIFFERENTIAL CATEGORIES 633 Proo. There are a number o things to check here are those that are not perectly obvious. Composition preserves additivity in the irst component: x + y, z F, π 1 G = x + y, z F, z G = x, z F + y, z F, z G = x, z F, z G + y, z F, z G = x, z F, π 1 G + y, z F, π 1 G Addition is well-deined (i.e. F + G is additive in the irst variable): x + y, z (F + G) = x + y, z F + x + y, z G (F + G, + g) is let additive: Products: Given = x, z F + x, z G + y, z F + y, z G = x, z (F + G) + y, z (F + G) (H, h)(f + G, + g) = ( H, π 1 h (F + G), h( + g)) (X, A) = ( H, π 1 h F + H, π 1 h G, h + hg) = ( H, π 1 h F, h) + ( H, π 1 h G, hg) = (H, h)(f, ) + (H, h)(g, g) (F,) (Z, C) (G,g) (Y, B) ( F,G,,g ) the induced map to the product is simply (Z, C) (X Y, A B). The point then is that this commutes as required, is unique, and inally that the projections are additive, and that pairing preserves additivity. One useul observation is that (F, ) is additive (in ABun(X)) i and only i both F, are additive (in X). ( F, G,, g )(π 0 π 0, π 0 ) = ( F, G, π 1, g π 0 π 0,, g π 0 ) = (F, ) and similarly or the other composite. Note that this also shows uniqueness, since the typing o the ill map (Z, C) (X Y, A B) orces it to be ( F, G,, g ), provided that commutes as required. The projectives are additive: ((F, ) + (G, g))(π 0 π 0, π 0 ) = (F + G, + g)(π 0 π 0, π 0 ) = ( F + G, π 1 ( + g) π 0 π 0, ( + g)π 0 ) = ((F + G)π 0, ( + g)π 0 ) = (F π 0 + Gπ 0, π 0 + gπ 0 ) = (F π 0, π 0 ) + (Gπ 0, gπ 0 ) = (F, )(π 0 π 0, π 0 ) + (G, g)(π 0 π 0, π 0 )

13 634 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY and similarly or π 1. Pairing preserves additivity: suppose (F, ), (G, g) are additive (so all F,, G, g are). Then ((H, h) + (K, k))( F, G,, g ) = (H + K, h + k)( F, G,, g ) = ( H + K, π 1 (h + k) F, G, (h + k), g ) = ( H + K, π 1 (h + k) F, H + K, π 1 (h + k) G, (h + k), (h + k)g ) = ( (H, π 1 h)f + (K, π 1 k)f, (H, π 1 h)g + (K, π 1 k)g, h + k, hg + kg ) = ( (H, π 1 h)f, (H, π 1 h)g, h, hg ) + ( (K, π 1 k)f, (K, π 1 k)g, k, kg ) = (H, h)( F, G,, g ) + (K, k)( F, G,, g ) Next we consider the projection unctor p: ABun(X) X which sends (X, A) A, (F, ) ; this is well-known to be a ibration, but there is more structure in our context: it is clear that p preserves, +, and so is a Cartesian let additive unctor. Hence: Proposition. I X is Cartesian let additive, then p: ABun(X) X is a Cartesian let additive unctor which is also a ibration, whose ibres are additive categories. We can axiomatize the essence o this structure: Deinition. p: Y X is a additive bundle ibration i it is a ibration satisying these properties. 1. X is let additive; 2. each ibre p 1 A is additive, or every object A o X; 3. : p 1 B p 1 A is additive, or every morphism A B o X; 4. there is an object unction (, ): Obj(X Y) Obj(Y) so that or any morphism A B o X, the Cartesian liting o to any object X over B is o the orm : (X, A) X, or in other words, X = (X, A), and does not depend on but merely on X and the domain o. Note that in this deinition we have not assumed that Y is let additive (even i rom ABun(X) we expect it to be), nor have we supposed that X and Y are Cartesian. The irst non-supposition turns out to be unnecessary, as the next Proposition shows. As or the second, we see that i X and the ibres are Cartesian, then so must Y be also. (This is amiliar territory rom ibrations, o course.) Proposition. I p: Y X is an additive bundle ibration, then 1. Y is let additive with respect to the additive bundle addition, and 2. i each p 1 A is Cartesian additive and i X is Cartesian let additive, then Y is Cartesian let additive as well.

14 CARTESIAN DIFFERENTIAL CATEGORIES 635 Proo. We shall use the ollowing notation: i : Y X is a morphism o Y over the morphism A p B in X, then its actorization into a ibre map ollowed by a Cartesian map over p will be written Y p (X, A) X. Then we may deine the additive structure as ollows: 0 = 0; 0, where we use 0 to denote the 0-map in the ibre over A, as well as the 0-map in Y and X, and +g = ( +g ); (p + pg). This is clearly commutative; urthermore 0 is a unit or +, and + is associative: + 0 = ( + 0)(p + 0) = ; p = ( + g) + h = ( + g + h )(p + pg + ph) = + (g + h) We need to show Y is let additive. (h + k) = ; (h + k ); (ph + pk) = ; p; (h + k ); (ph + pk) = ; ( h + k ); p; (ph + pk) = ( h + k ); (p(h) + p(k)) = ((h) + (k) ); (p(h) + p(k)) (by uniqueness o actorization) = h + k 0 = ( p)(00) = 0(p)(p0) = 0(p0) = 00 Now, we suppose that the base category and the ibres are Cartesian; note that each is additive, so preserves products. Products in Y are deined in the standard manner, pulling back to a common ibre and orming the product there. We must show that the projections are additive, and that the pair o additive maps is additive. We shall ind the ollowing lemma useul here Lemma. Any map in Y is additive i and only i and p are additive, in the ollowing sense: or suitably typed maps h, k, the ollowing equations hold: ((h + k)) = (h) + (k) and p((h + k)) = p(h + k). Proo. (o the lemma) The ibre-map actor o (h + k) is (h + k )(ph + pk) = ((h + k)), and the Cartesian actor is (ph + pk)p = (ph + pk)p. The ibre-map actor o h+k is h (ph) +k (pk) = (h) +(k), and the Cartesian actor is p(h) + p(k). The lemma is obvious rom these actorizations.

15 636 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY So obviously the projections are additive; additionally, g is additive i, g are. ( h, hg + k, kg ) = h, hg + k, kg = (h), (hg) + (k), (kg) = (h) + (k), (hg) + (kg) = ((h + k)), ((h + k)g) p( h, hg + k, kg ) = p(h), p(hg) + p(k), p(kg) = p(h) + p(k), p(hg) + p(kg) = p(h + k), p(hg + kg) = p((h + k)), p((h + k)g) We have essentially already shown that ABun(X) X is an additive bundle ibration; Proposition may be restated as ollows: Proposition. I X is let additive, then p: ABun(X) X is an additive bundle ibration. Furthermore, i X is Cartesian let additive, then p: ABun(X) X is a Cartesian let additive unctor. 2. Cartesian dierential categories Having developed the structure o let additive categories we are now ready to introduce the notion o a Cartesian dierential category. Fundamental to these categories is the (appropriate) notion o a dierential operator. Consider the category o inite dimensional vector spaces over (or example) the reals, with homomorphisms which are ininitely dierentiable maps. This is let additive, it has products, and urthermore has a natural dierential operator given by the Jacobian. For example, consider the map (x 1, x 2 ) = x 2 1 +x 2 2: IR 2 IR. Its Jacobian is [ ] 2x 1 2x 2. Note that the Jacobian produces rom the point (x 1, x 2 ) a linear map rom IR 2 IR, and so ( uncurrying ) we get D (): IR 2 IR 2 IR. It is this property that we shall abstract with the notion o a Cartesian dierential operator The basic deinitions Deinition. An operator D on the maps o a Cartesian let additive category X Y X X D [] Y is a Cartesian dierential operator in case it satisies the ollowing: [CD.1] D [ + g] = D [] + D [g] and D [0] = 0

16 CARTESIAN DIFFERENTIAL CATEGORIES 637 [CD.2] h + k, v D [] = h, v D [] + k, v D [] and 0, v D [] = 0 [CD.3] D [1] = π 0, D [π 0 ] = π 0 π 0 and D [π 1 ] = π 0 π 1 [CD.4] D [, g ] = D [], D [g] [CD.5] D [g] = D [], π 1 D [g] [CD.6] g, 0, h, k D [D []] = g, k D [] [CD.7] 0, h, g, k D [D []] = 0, g, h, k D [D []] Note that the nullary case o [CD.4], D [ ] =, automatically holds, since 1 is terminal. [CD.6] may equivalently be stated as ( 1, 0 1)D [D []] = (1 π 1 )D [] Somewhat less obviously (but rather crucially or what ollows): Lemma. [CD.7] is equivalent to [CD.7 ]: 0, 0, h, 0, 0, g, k 1, k 2 D [D []] = 0, 0, 0, g, h, 0, k 1, k 2 D [D []] Proo. It is clear that this is implied by [CD.7]; to establish the converse, consider that by [CD.7 ] 0, h, 0, 0, g, 0, k D[D[(π 0 + π 1 )]] = 0, 0, g, h, 0, 0, k D[D[(π 0 + π 1 )]] But D[D[(π 0 + π 1 )]] = ((π 0 + π 1 ) (π 0 + π 1 )) ((π 0 + π 1 ) (π 0 + π 1 ))D[D[]] since (anticipating Deinition and Lemma 2.2.2) (π 0 + π 1 ) is linear, and so we obtain [CD.7] Remark. Some comments on these axioms might help the reader with their meanings. [CD.1] says D is linear; [CD.2] that it is additive in its irst coordinate. [CD.3,4] assert that D behaves coherently with the product structure, and [CD.5] is the chain rule. We shall see rom the proo o Lemma (v) that [CD.6] is essentially requiring that D [] be linear (in the sense o Deinition 2.2.1) in its irst variable (more precisely, that 1, 0 D [] is linear). [CD.7] will be clearer when we restate it with a term logic: at that time it will be clear that [CD.7] is just independence o order o partial dierentiation : 2 = 2. x y y x

17 638 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY Deinition. A Cartesian let additive category which has a Cartesian dierential operator is a Cartesian dierential category. The category o inite dimensional vector spaces with smooth maps is an example. 2 Here we can deine a smooth dierential operator via the Jacobi matrix: where x = (x 1,..., x m ) and ỹ = (y 1,.., y m ). D s [( 1,.., n )]( x, ỹ) = [( xi j )(ỹ)] m,n i=1,j=1 x 2.2. The subcategory o linear maps. In a Cartesian dierential category, not only is there a subcategory o additive maps but there is also a subcategory o maps whose dierential is constant, that is maps which are linear : Deinition. A map in a Cartesian dierential category is said to be linear in case D [] = π 0. The ollowing lemma establishes the basic properties o the linear maps in a Cartesian dierential category Lemma. In any Cartesian dierential category (i) Every linear map is additive; (ii) 0 is a linear map, and i and g are linear then + g is linear; (iii) Linear maps compose, and include the identity maps; (iv) Projections are linear and pairings o linear maps are linear; (v) 1, 0 D [] is linear; (vi) I a and b are linear and i the lethand square commutes then the righthand square also commutes. A B a b = A B A A a a D [] A A D [ ] B b B (vii) I g is a retraction and linear and gh is linear then h is linear; (viii) I is an isomorphism and linear then 1 is linear. 2 Indeed this is true o any dierential theory over a rig [BCS 06], and so this gives a large class o examples.

18 CARTESIAN DIFFERENTIAL CATEGORIES 639 Proo. (i) Consider ( + g)h where D [h] = π 0 h then ( + g)h = + g, D [h] =, D [h] + g, D [h] = h + gh. (ii) D [0] = 0 = π 0 0 so the zero map is linear. When and g are linear then D [ + g] = D [] + D [g] = π 0 + π 0 g = π 0 ( + g). (iii) Identity maps by deinition are linear. Suppose and g are linear, then D [g] = D [], π 1 D [g] = π 0, π 1 π 0 g = π 0 g. (iv) Projections are linear by deinition. The pairing o two linear maps is linear as D [, g ] = D [], D [g] = π 0, π 0 g = π 0, g. (v) We have the ollowing calculation: D [ 1, 0 D []] = D [ 1, 0 ], π 1 1, 0 D [D []] = π 0 1, 0, π 1 1, 0 D [D []] = (1 1, 0 )( 1, 0 1)D [D []] = (1 1, 0 )(1 π 1 )D [] = π 0 1, 0 D []. (vi) I a = b then D [a ] = D [b] but now D [a ] = D [a], π 1 a D [ ] = π 0 a, π 1 a D [ ] = (a a)d [ ] D [b] = D [], π 1 D [b] = D [], π 1 π 0 b = D []b showing that the above inerence holds. (vii) Let g g = 1 we need to determine the value o D [h] or this we have: D [h] = (g g )(g g)d [h] = (g g ) π 0 g, π 1 g D [h] = (g g ) D [g], π 1 g D [h] = (g g )D[gh] = (g g )π 0 gh = π 0 g gh = π 0 h. (viii) This ollows easily rom the property above.

19 640 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY From Lemma we may conclude: Corollary. The linear maps o a Cartesian dierential category orm an additive category X lin which has biproducts. The inclusion I: X lin X relects isomorphisms and creates products An additive interlude!. In a Cartesian dierential category the additive maps play second iddle to the linear maps. Nonetheless they are an important class which have many o the properties o the linear maps. Below we develop some o the special properties o additive maps beore turning our attention to the linear maps. As a consequence (Corollary 2.3.3), we shall also show that axiom [CD.6] is independent o the other axioms. (A proo that axiom [CD.7] is independent o the other axioms may be constructed rom a modiication o the construction o the ree Cartesian dierential category; that will appear in a sequel which develops the technical tools needed or that construction.) Proposition. In a Cartesian dierential category i is additive then D [] is additive and, urthermore, D [] = π 0 D 0 [] where D 0 [] = 1, 0 D []. Proo. Suppose is additive so that π 0 + π 1 = (π 0 + π 1 ) then π 0 D [] + π 1 D [] = ex( π 0 π 0, π 1 π 0 D [] + π 0 π 1, π 1 π 1 D []) = ex( D [π 0 ], π 1 π 0 D [] + D [π 1 ], π 1 π 1 D []) = ex(d [π 0 ] + D [π 1 ]) = exd [π 0 + π 1 ] = exd [(π 0 + π 1 )] = ex D [π 0 + π 1 ], π 1 (π 0 + π 1 ) D [] = ex π 0 (π 0 + π 1 ), π 1 (π 0 + π 1 ) D [] = (π 0 + π 1 )D [] where ex = π 0 π 0, π 1 π 0, π 0 π 1, π 1 π 1 is the exchange map. Now, when is additive, we can use the act D [] is additive to conclude: D [] = π 0, π 1 D [] = 0 + π 0, π D [] = ( 0, π 1 + π 0, 0 )D [] = 0, π 1 D [] + π 0, 0 D [] = π 0, 0 D [] = π 0 1, 0 D [] so that D [] = π 0 1, 0 D [] = π 0 D 0 []. In the extremal situation when every map is additive we may now conclude that the dierential D [] = π 0 D 0 [] is given by D 0 which is an additive endounctor stationary on objects and linear maps and delivers linear maps and so is idempotent.

20 CARTESIAN DIFFERENTIAL CATEGORIES Corollary. In a Cartesian dierential category X in which all maps are additive D 0 : X X is an additive endounctor which is stationary on all objects and linear maps and has image the linear maps. Conversely, an endounctor D 0 which is stationary on objects and idempotent endows such a category with a dierential operator D [] = π 0 D 0 []. Proo. I is linear then D 0 [] = 1, 0 D [] = 1, 0 π 0 =. As D 0 [+g] = 1, 0 D [+g] = 1, 0 (D []+D [g]) = 1, 0 D []+ 1, 0 D [g] = D 0 []+D 0 [g] so that D 0 preserves addition, it also clearly preserves the zero. D 0 preserves composition as: D 0 [g] = 1, 0 D [g] = 1, 0 D [], π 1 D [g] = 1, 0 D [], 0 D [g] = D 0 []D 0 [g]. For the converse suppose D 0 : X X is a unctor which is additive, stationary on objects, and has is idempotent. It preserves the biproduct structure as all additive unctors do. Now deine D [] = π 0 D 0 () then it is easy to check that this is a Cartesian dierential operator. These observations yield some unexpected examples o Cartesian dierential categories. First consider the category o matrices over the complex numbers (the objects being natural numbers and the maps being matrices) then there is an obvious endounctor which takes the complex conjugate o each matrix entry. This leaves the product structure untouched. This makes it a (non-trivial) additive endo unctor which however is not idempotent. By treating this as the D 0 o corollary we get a structure D [] = π 0 D 0 [] which satisies all the equations except [CD.6]. This means: Corollary. [CD.6] is independent o the rest o the axioms. To get an example o a structure which also satisies this axiom it suices to consider a ring with a retraction: or example the polynomial ring over the complex numbers retract onto the complexes by assigning the indeterminate to any number: r x:=0 : C[x] C. This extends to an additive idempotent endounctor on the category o the matrices over C[x] which can serve as the endounctor D 0 in the above. This makes the matrices with entries in C the linear maps and the rest have a dierential which collapses back to C. Clearly, an interesting case is when the linear maps and additive maps coincide. One may express this by the requirement that D 0 is the identity unctor on the additive maps. The example above shows that this, i desired, is an extra requirement Corollary. There are Cartesian dierential categories where the additive maps are not all linear. It seems appropriate to make one urther remark about the operator D 0 [] = 1, 0 D []. One should think o it as giving the dierential at 0. O course, when one ixes the point

21 642 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY at which one takes the dierential one obtains an additive map. Thereore, let X 0 be the subcategory o X determined by those maps which preserve the zero (0 = 0). Clearly any D [] preserves the 0 by [CD.2], thus X 0 is itsel a Cartesian dierential subcategory which will usually strictly include all the additive maps Corollary. I X is a Cartesian dierential category then X 0, consisting o the maps which preserve 0 (i.e. satisying 0 = 0), is a Cartesian dierential subcategory. A urther class o maps should be mentioned namely the constant maps, that is those with dierential zero. Classically they are important as the dierential o a map will not be changed by adding constant maps to it Cartesian dierential operators and the bundle ibration. Recall (Section 1.5) that i X is Cartesian let additive, then p: ABun(X) X is an additive bundle ibration, and p: ABun(X) X is a Cartesian let additive unctor. Suppose p: ABun(X) X has some urther structure: suppose p has a let additive section D: X ABun(X). Some interesting consequences ollow rom this supposition, consequences which one may take as motivation or the axioms o a Cartesian dierential operator. First, some notation: let s write D(A) = (d 0 (A), A) and D() = (D [], ), or A an object o X, and : A B a map o X. (The assumption that D is a section orces the second component to be as given.) Note then that D []: d 0 (A) A d 0 (B) is additive in its irst component. Also, that D is a unctor orces the ollowing equation: D [], π 1 D [g] = D [g] because the lethand side is the irst component o D()D(g) and the righthand side is the irst component o D(g). (In each case the second component is just g.) Our view will be that ABun(X) captures dierential structure o X, and that composition in ABun(X) should then be governed by the chain rule this is exactly what the equation above expresses. In addition, that D preserves identities means that 1 = (π 0, 1) = (D [1], 1), and so Also D(π 0 ) = (π 0 π 0, π 0 ), so Similarly D [1] = π 0 D [π 0 ] = π 0 π 0 D [π 1 ] = π 0 π 1 D preserves +: D( + g) = D() + D(g), and so D [ + g] = D [] + D [g] So what we see here is that part o the deinition o Cartesian dierential operator (Deinition 2.1.1), speciically [CD.1-5], ollows immediately rom the existence o a let additive section D to p.

22 CARTESIAN DIFFERENTIAL CATEGORIES 643 But we can in act push this analysis o the additive bundle ibration urther: we can deine the notion o linear map in this context. We say a map o X is linear i D [] = π 0 g or some g. Then we note that the basic properties o linear maps hold in this generality, with more or less the same proos. g is uniquely determined by (since π 0 : d 0 A A 1, 0 ). We shall denote the map g by d 0 (). d 0 A is epi, having a section d 0 so deined is an endo-unctor on X, and linear maps orm an additive subcategory o X. D [1] = π 0, so identities are linear; i D [] = π 0 d 0 and D [g] = π 0 d 0 g, then D [g] = D [], π 1 D [g] = π 0 0, π 1 π 0 d 0 g = π 0 d 0 d 0 g, so linear maps are closed under composition; i, g are linear, then D [ + g] = D [] + D [g] = π 0 d 0 + π 0 d + 0g = π 0 (d 0 + d 0 g), so + g is linear as well. I is additive, then so is d 0 : (x + y)d 0 = x + y, 0 π 0 d 0 = x + y, 0 D [] = x, 0 D [] + y, 0 D [] = x, 0 π 0 d 0 + y, 0 π 0 d 0 = xd 0 + yd 0 Projections are linear; pairs o linear maps are linear. The special case which concerns us is when the unctor d 0 is the identity (we suggest calling D stationary in this case). When that is so, then we have the rest o the basic properties: I a and b are linear and i the lethand square commutes then the righthand square also commutes. A B a b = A B A A a a D [] A A D [ ] B b B I g is a retraction and linear and gh is linear then h is linear. Hence, i is an isomorphism and linear then 1 is linear.

23 644 R.F. BLUTE, J.R.B. COCKETT AND R.A.G. SEELY A proper analysis in this ashion o [CD.6,7] involves higher order dierential structure: just as the chain rule essentially characterizes composition in additive bundle ibrations, the higher order chain rules induce a amily o such ibrations, the Faà di Bruno ibrations, corresponding to higher order dierential structure a well-behaved composition in the second order Faà di Bruno ibration will motivate [CD.6,7] in a similar manner (the details are somewhat technical, and will be presented in a sequel). 3. The cokleisli category o a dierential category First, we shall recall some deinitions rom [BCS 06], and set our notation Dierential categories. A coalgebra modality on an additive (i.e. commutative monoid enriched) symmetric monoidal category X is a comonad (S, δ, ) so that each object S(X) is a coalgebra or comonoid (S(X),, e) and δ is a comonoid morphism. : S(X) S(X) S(X) e: S(X) S(X) S(X) S(X) S(X) S(X) 1 e e 1 δ S(X) S(S(X)) e e S(X) S(X) S(X) 1 S(X) S(X) 1 S(X) S(X) S(X) S(X) S(X) S(X) δ δ δ S(S(X)) S(S(X)) S(S(X)) (We wrote! instead o S in [BCS 06]; S will it better with the rest o this paper.) Note that we do not require X to have biproducts. On such a category, a dierential combinator D : X(S(A), B) X(A S(A), B) is a natural combinator satisying our axioms (we suppress structural isomorphisms): [D.1] (Constant maps) D [e A ] = 0 [D.2] (Product rule) D [( g)] = (1 )(D [] g) + (1 )(c 1)( D [g]), where : S(A) B, g: S(A) C, and c is the commutativity isomorphism [D.3] (Linearity) D [ A ] = (1 e A ), where : A B [D.4] (Chain rule) D [δ S() g] = (1 )(D [] δs())d [g] where : S(A) and g: S(B) C. B A dierential category is an additive symmetric monoidal category with a coalgebra modality and a dierential combinator. The structure o the dierential combinator may also be described in slightly simpler terms using a deriving transormation d : = D [1].

24 CARTESIAN DIFFERENTIAL CATEGORIES 645 (The our axioms have a slightly simpler ormulation in terms o d.) In [BCS 06] we used the notation D and d; in this paper we also have similar Cartesian operations D, d or Cartesian dierential categories, so we shall use the subscripted D, d to distinguish the dierential combinator and deriving transormation o a dierential category rom their counterparts in a Cartesian dierential category. In [BCS 06], we also deine several richer variants o dierential categories: or example, i X has biproducts, we deine a dierential storage category as an additive storage category with a deriving transormation such that the -rule (d 1) = (1 )d is satisied, ( being the codiagonal, part o the bialgebra structure). Being a storage category means that the coalgebra modality is symmetric monoidal, and the comonoid structure is a morphism or the coalgebras. This is equivalent to the induced tensor on the coalgebras being a product, and in act in this setting we obtain the storage (or Seely) isomorphisms which mediate a canonical bialgebra structure on the coree objects by transporting the bialgebra structure on the biproduct. The point about dierential storage categories rom our present viewpoint is that the deriving transormation is given by the bialgebra structure: d X = (η X 1), or a natural map η. The reader should consult [BCS 06] or the details o this and the other variants The cokleisli category o a dierential category. Our principle example o a Cartesian dierential category arises as the cokleisli category o a dierential category. In act, under reasonable conditions, we may represent Cartesian dierential categories as the cokleisli categories o dierential categories; in order to do that, we need to consider abstract cokleisli categories in general, which will be the main ocus o a sequel to this paper. But or now, let us consider this example o Cartesian dierential categories. As beore, to distinguish maps in the cokleisli category, we shall represent them in boldace. The key this example is that, given a dierential category X with products (and so biproducts), it is possible to deine a dierential combinator on the cokleisli category: given X Y in the cokleisli category (and so S(X) Y in the underlying category), we construct D []: X X Y in the cokleisli category as the arrow (in the underlying category) D [] = S(X X) S(X X) S(X X) Sπ 0 Sπ 1 S(X) S(X) 1 X S(X) D [] Y We remark that the arrow s 2 : = S(X X) S(X X) S(X X) Sπ 0 Sπ 1 S(X) S(X) is the canonical (inverse) storage isomorphism, in a category which has the storage isomorphism or example, or a storage modality. However, we are not supposing this is an isomorphism here. In act, having s (and its nullary version) amounts to having a coalgebra modality.