Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base

Transcription

1 Under consideration or publication in J. Functional Programming 1 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base STEPHEN LACK School o Computing and Mathematics, University o Western Sydney, Locked Bag 1797, Penrith South DC, NSW 1797, Australia ( s.lack@uws.edu.au) JOHN POWER Department o Computer Science, University o Bath, Claverton Down, Bath BA2 7AY, United Kingdom ( ajp@in.ed.ac.uk) Abstract Motivated by the search or a body o mathematical theory to support the semantics o computational eects, we irst recall the relationship between Lawvere theories and monads on Set. We generalise that relationship rom Set to an arbitrary locally presentable category such as P oset, ωcpo, or unctor categories such as [Inj, Set] or [Inj, ωcpo]. That involves allowing the arities o Lawvere theories to be extended to being size-restricted objects o the locally presentable category. We develop a body o theory at this level o generality, in particular explaining how the relationship between generalised Lawvere theories and monads extends Gabriel-Ulmer duality. 1 Introduction Over the twenty years since Eugenio Moggi wrote the seminal papers (Moggi, 1989; Moggi, 1991), the notion o monad has become a valuable tool in the study o unctional programming languages, both or call-by-value languages like M L and or call-by-name or call-by-need languages like Haskell (Benton et al., 2000), speciically in regard to the modelling o computational eects. Over the past ten years, substantial progress has been made, especially in regard to the theoretical study o combining eects, by observing that almost all monads o computational interest on Set arise naturally rom countable Lawvere theories (Plotkin and Power, The support o the ARC and DETYA is grateully acknowledged. This work has been done with the support o EPSRC grant GR/586372/01 and a visit to AIST, Senri-Chou, Japan.

2 2 Stephen Lack and John Power 2002; Hyland et al., 2006; Hyland and Power, 2006; Hyland et al., 2007), with the combinations o eects determined principally by the sum or tensor, sometimes the distributive tensor, o the corresponding Lawvere theories. We recall the deinition and leading examples in Section 2. Lawvere theories are a category-theoretic ormulation o universal algebra or which the notion o operation is primitive (Lawvere, 1963). The deinition o Lawvere theory axiomatises the notion o the clone o an equational theory. Unlike equational theories, Lawvere theories are presentation-independent, i.e., the category o models determines a Lawvere theory uniquely up to coherent isomorphism. Every Lawvere theory generates a monad on Set, generating precisely the initary monads on Set. The relationship between Lawvere theories, equational theories and initary monads on Set is one o the deepest relationships in category theory (Hyland and Power, 2007). But Set is not the base category o primary interest or computation: the category ωcpo is more interesting as it incorporates an account o recursion. The deinition o monad extends routinely rom base category Set to an arbitrary base category, hence to ωcpo, but the deinition o Lawvere theory does not. To some extent, that can be resolved by appeal to the notion o an enriched Lawvere theory (Power, 2000), as was used in (Hyland et al., 2006; Hyland and Power, 2006; Hyland et al., 2007). But enrichment is less appropriate when one wants to replace Set by categories such as the unctor categories [C, Set] or [C, ωcpo], or instance in order to model local eects, notably local state, as investigated in (O Hearn and Tennent, 1997; Plotkin and Power, 2002; Power, 2006). So we seek a generalisation o the deinition o Lawvere theory that applies to categories such as those cited above, with the relationship with monads respected by the generalisation. A start on that question was made in the mathematical paper (Nishizawa and Power, 2007), and in this paper, we develop it urther and explain it in a computational setting. For ease o exposition, we shall ignore enrichment beyond saying that everything we write enriches without uss, using the techniques o (Kelly, 1982; Nishizawa and Power, 2007), explained in the setting o computational eects in (Hyland et al., 2006). We provide reerences through the paper to background material expressed in the enriched setting. The mathematical oundation o the paper is Gabriel-Ulmer duality. The axiom we require o a base category is that it is locally initely presentable, or, slightly more generally, locally countably presentable. Gabriel-Ulmer duality asserts that every locally initely presentable category A is equivalent to the category F L(A op, Set) o inite limit preserving unctors rom A op to Set, where A is determined by what are called the initely presentable objects o A: these generalise the notion o inite set. The situation or countability is similar. We outline the main ideas o Gabriel-Ulmer duality in Section 3. Based upon Gabriel-Ulmer duality, in Section 4, we recall the deinition o Lawvere A-theory or a locally initely presentable category A rom (Nishizawa and Power, 2007) and give a new deinition o model, which we prove equivalent to that o (Nishizawa and Power, 2007): the latter is more directly applicable to examples, but the ormer allows or a more elegant explanation o the relationship with

3 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 3 monads as supported by Gabriel-Ulmer duality. We explain the relationship between our deinitions and the building blocks o Gabriel-Ulmer duality in Section 5. The central result o the paper, in Section 6, is Corollary 23, which expresses the equivalence between Lawvere A-theories and initary monads on A as a liting o Gabriel-Ulmer duality over A. We extend Gabriel-Ulmer duality to examine change o base in Section 7: one seeks not only a characterisation o the monads with countable rank on a category such as ωcpo, but also a relationship between such monads on ωcpo, equivalently countable Lawvere ωcpo-theories, and monads with countable rank on Set, equivalently ordinary countable Lawvere theories. 2 Ordinary Lawvere Theories In this section, we recall the notion o Lawvere theory, irst deined in Lawvere s thesis (see (Lawvere, 1963)), its relationship with monads on Set, and its relevance to unctional programming with computational eects. The examples are taken rom (Hyland et al., 2006), which in turn was motivated by the desire to reine and develop Moggi s modelling o computational eects by monads in (Moggi, 1989; Moggi, 1991). Deinition 1 A Lawvere theory consists o a small category L with inite products together with a strict inite product preserving identity-on-objects unctor I : Nat op L, where Nat is the category o all natural numbers and maps between them (Barr and Wells, 1985; Barr and Wells, 1990). A model o a Lawvere theory L in a category C with inite products is a inite-product preserving unctor rom L to C. Implicit in Deinition 1 is the act that the objects o L are exactly the natural numbers. A map o Lawvere theories rom I : Nat op L to I : Nat op L is a unctor rom L to L that respects I. Any such unctor is necessarily strictly inite product preserving and identity-on-objects. With the usual composition o unctors, this yields a category Law. Note the distinction in the deinition between strict preservation, as used in deining a Lawvere theory, and preservation, as used in deining a model. The latter means that inite products need only be preserved up to coherent isomorphism rather than equality. The distinction is essential as, on one hand, the objects o L are exactly the natural numbers, but on the other, i we demanded strict preservation in the deinition o model, we would eliminate almost all examples o interest (Power, 1995). The deinition o Lawvere theory provides a category theoretic ormulation o universal algebra, with the notion o operation taken as primitive: a map in L rom n to m is to be understood as being given by m operations o arity n. Unlike the notion o equational theory, the concept o Lawvere theory is presentationindependent, i.e., i a pair o Lawvere theories have equivalent categories o models, the two theories are isomorphic.

4 4 Stephen Lack and John Power The deinition o model extends to the deinition o the category Mod(L, C) o models o L in C: maps o models are deined to be natural transormations. Note that naturality orces maps o models to respect the inite product structure in the deinition o model. For any Lawvere theory L and any locally initely presentable category C (characterised in Section 3), the unctor ev 1 : Mod(L, C) C has a let adjoint, given by the ree model construction, inducing a monad T L on C. Thus in particular, every Lawvere theory L determines a monad T L on Set. There is a converse construction. Proposition 2 Given any monad T on Set, i one actorises the canonical composite Nat Set Kl(T ) where Kl(T ) denotes the Kleisli category o T, as an identity-on-objects unctor ollowed by a ully aithul unctor, one obtains (the opposite o) a Lawvere theory I : Nat L op T Proo By construction, the unctor I is identity-on-objects. Moreover, the canonical unctor Set Kl(T ) strictly preserves colimits, in particular all coproducts. Restricting to N at and actorising as above ensures that I strictly preserves all inite coproducts. Applying ( ) op, we are done. I one started with a Lawvere theory L, constructed T L, and then constructed L TL, one would recover L. But the converse is not true: starting with a monad T, one does not in general recover T or size reasons: one recovers T i and only i T is initary, i.e., i and only i T preserves iltered colimits: these are a special orm o colimit, or which a precise understanding is not essential here. Putting this together, with care or coherence detail, yields the ollowing (Power, 2000). Theorem 3 The constructions o a monad T L on Set rom a Lawvere theory L together with that o a Lawvere theory L T rom a monad T on Set extend canonically to an equivalence o categories Law Mnd where Mnd is the category o initary monads on Set. Moreover, or any Lawvere theory L, the category Mod(L, Set) is coherently equivalent to T L -Alg. The usual way in which one obtains Lawvere theories is by means o sketches, or equivalently, equational theories, with the Lawvere theory given reely on the sketch: (Barr and Wells, 1990) treats sketches in loving detail and gives a range o examples o both sketches and Lawvere theories. The Lawvere theory is an axiomatisation o the notion o the clone o an equational theory, equivalently o a sketch.

5 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 5 Example 4 The Lawvere theory L E or exceptions is the ree Lawvere theory generated by an E-indexed amily o nullary operations with no equations. The monad on Set induced by L E is T E = + E. More generally, i C is any category with inite products and all sums, then Mod(L E, C) is equivalent to the category o algebras or the monad + E where E is the E-old coproduct o copies o 1, i.e., E 1. Interactive input/output works similarly to exceptions (Hyland et al., 2006), so we omit details. For the next example, we use the evident generalisation o the notion o Lawvere theory to countable Lawvere theory as used in (Hyland et al., 2006): it simply allows us to use countable arities. Example 5 Let Loc be a inite set o locations and let V be a countable set o values. The countable Lawvere theory L GS or side-eects, sometimes called global state, where S = V Loc, is the ree countable Lawvere theory generated by the operations lookup: V Loc and update: 1 Loc V subject to the seven natural equations listed in (Plotkin and Power, 2002), our o them speciying interaction equations or lookup and update and three o them speciying commutation equations. Our presentation o the operations here is in terms o generic eects, corresponding to the evident unctions o the orm Loc (S V ) S and Loc V S S respectively (Hyland et al., 2006): to give a generic eect is equivalent, via the Yoneda embedding, to giving an operation (Plotkin and Power, 2003). It is shown in (Plotkin and Power, 2002) that L GS induces the side-eects monad. More generally, i C is any category with countable products and copowers then, slightly generalising the result in (Plotkin and Power, 2002), Mod(L GS, C) is equivalent to the category o algebras or the monad (S ) S where we write (S ) or the S-old coproduct S, and ( )S or the S-old product S. Example 6 The Lawvere theory L N or (binary) nondeterminism is the Lawvere theory reely generated by a binary operation : 2 1 subject to equations or associativity, commutativity and idempotence, i.e., the Lawvere theory or a semilattice. The induced monad on Set is the inite non-empty subset monad, F +. Example 7 The Lawvere theory L P or probabilistic nondeterminism is that reely generated by [0, 1]-many binary operations + r :2 1 subject to natural equations generalising associativity, commutativity and idempotence (Heckmann, 1994). The induced monad on Set is the distributions with inite support monad, D. For a non-example, consider the monad ( ) on P oset or ωcpo or the addition o a least element. The monad ( ) does not arise rom an ordinary Lawvere theory as one cannot express as an equation the assertion that or all x, one has x. So one needs to go beyond ordinary Lawvere theories in order to include such monads on categories such as P oset or ωcpo. In (Hyland et al., 2006), enriched Lawvere theories were used, with enrichment in P oset or ωcpo respectively. In this paper, we

6 6 Stephen Lack and John Power propose a dierent generalisation that works similarly well or ( ) and better than enriched Lawvere theories or base categories such as unctor categories [C, Set] or [C, ωcpo] as used to model local state (O Hearn and Tennent, 1997; Plotkin and Power, 2002; Power, 2006). 3 Gabriel-Ulmer Duality The deinition o a Lawvere theory, Deinition 1, involves the category Nat o natural numbers, with natural numbers orming the possible arities o an operation. So, i we are to axiomatise the deinition, we need to be able to speak meaningully o the inite objects o a category A, as the inite objects o A are the possible arities or an A-based Lawvere theory. That problem was deinitively resolved several decades ago by the notion o a locally initely presentable category and the theory o Gabriel-Ulmer duality (Adámek and Rosický, 1994; Kelly, 1982a): the appropriate objects are called the initely presentable objects o A. The deinition o a locally initely presentable category is quite complex (Adámek and Rosický, 1994). But the central result o Gabriel-Ulmer duality characterises the notion in simple terms as ollows (Adámek and Rosický, 1994; Kelly, 1982a). Let FL denote the 2-category o all small categories with inite limits, inite limit preserving unctors, and all natural transormations. Given a category C with inite limits, let F L(C, Set) denote the ull subcategory o the unctor category [C, Set] determined by those unctors that preserve inite limits. And let LocPres denote the 2-category o locally initely presentable categories, iltered colimit preserving unctors that have let adjoints, and natural transormations. Theorem 8 (Gabriel-Ulmer Duality) The construction that sends a small category C with inite limits to the category F L(C, Set) extends canonically to a biequivalence o 2-categories FL LocPres op So the study o locally initely presentable categories is equivalent to the study o categories o the orm F L(C, Set) where C is a small category with inite limits. Examples o locally initely presentable categories in the computer science literature include Set, Set k, P oset, Cat, and all unctor categories [C, A] or which C is a small category and A is a locally initely presentable category (Barr and Wells, 1990; Robinson, 2002). Gabriel-Ulmer duality extends routinely rom initeness to countability, allowing examples to include ωcpo and unctor categories o the orm [C, ωcpo], hence the categories o primary interest or recursion and local eects. Papers such as (Hyland et al., 2006) were written primarily in terms o countability, whereas the relevant mathematical literature is usually phrased in terms o initeness. The converse construction or Theorem 8 sends a locally initely presentable category A to a skeleton o the opposite o the ull subcategory o initely presentable objects o A. We shall duly write A or a skeleton o the ull subcategory o A given by the initely presentable objects o A and let ι : A A denote the inclusion

7 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 7 unctor. For example, the initely presentable objects o Set are the inite sets, and so Set is Nat. The initely presentable objects o P oset are the inite posets, and so P oset contains one isomorphic copy o each inite poset, with maps given by all maps o posets. Extending to the countable setting, the countably presentable objects o ωcpo include all countable ω-cpo s but also include uncountable ω-cpo s that have a countable presentation. For more detail in the computing literature, see (Robinson, 2002). In practice, one almost only ever needs to know some o the countably presentable objects o a locally countably presentable category, e.g., knowing that the countable ω-cpo s are among the countably presentable ones. A central act about A is as ollows. Proposition 9 For any locally initely presentable category A, the category A has all inite colimits and they are preserved by the inclusion ι : A A. We denote the composite unctor A Y [A op, Set] [ιop, Set] [A op, Set] by ι, where Y is the Yoneda embedding. For example, Set is Nat, and the unctor ι sends a set X to the unctor Set(ι, X), i.e., to X ( ). Since ι preserves all inite colimits, and representable unctors preserve limits, ι actors through F L(A op to F L(A op ollowing., Set). So we sometimes consider ι as a unctor rom A, Set). Unwinding the converse construction or Theorem 8, one has the Theorem 10 For any locally initely presentable category A, the unctor ι induces an equivalence o categories A F L(A op, Set) We shall return to Theorem 10 when we discuss models. 4 Lawvere A-theories In generalising Deinition 1, a tentative deinition o an A-based Lawvere theory might be a small category L with inite products together with a strict inite product preserving identity-on-objects unctor I : A op L. But such a deinition would not be delicate enough to allow us to generalise the relationship between Lawvere theories and monads as the ollowing example illustrates. Example 11 Let A = P oset. The category P oset is (equivalent to) the ull subcategory o P oset determined by the inite posets. Given a monad T on P oset, the canonical composite P oset P oset Kl(T ) preserves all inite colimits, and so the restriction I : P oset L op T

8 8 Stephen Lack and John Power strictly preserves all inite colimits. But that is a strictly stronger condition than that o strict preservation o inite coproducts. For consider the ollowing pushout in P oset : where 2 denotes the poset with two elements, with one less than the other, i.e., Sierpinski space, 3 is similar but with three elements, and with the evident maps between the various posets. Preservation o this pushout is not implied by preservation o inite coproducts, but i we are to axiomatise the relationship between Lawvere theories and monads so that it extends to P oset, this pushout must be preserved in the deinition o a P oset-based Lawvere theory as every P oset-based Lawvere theory must arise rom a monad on P oset. Guided by this example and relying upon Proposition 9, we make the ollowing deinition. The deinition o Lawvere A-theory we give here is identical to that given in (Nishizawa and Power, 2007), but the deinition o model we give here does not, a priori, agree with that given in (Nishizawa and Power, 2007): later, we shall prove that the two deinitions o model agree up to coherent isomorphism. Deinition 12 Given a locally initely presentable category A, a Lawvere A-theory is a small category L together with a strict inite limit preserving identity-on-objects unctor I: A op L. A model o a Lawvere A-theory I : A op L is a unctor M : L Set or which the composite MI preserves inite limits. The restriction o models to be Set-valued unctors in Deinition 12 while models were taken in any category C with inite products in Deinition 1 is essentially a convenience or exposition. We discuss the general situation at the end o the paper. A map o Lawvere A-theories rom L to L is an identity-on-objects unctor rom L to L that commutes with the unctors rom A op. Together with the usual composition o unctors, Lawvere A-theories and their maps yield a category we denote by Law A. The deinition o model routinely extends to the deinition o the category M od(l) o models o L, and Theorem 10 induces a canonical unctor U L : Mod(L) A Compare Deinition 1 with Deinition 12: the deinition o ordinary Lawvere theory required that L have inite products and that the unctor rom Nat op to L strictly preserve inite products, whereas here, we have asked or strict preservation o inite limits but have made no urther assumption o existence o any kind o limits in L. So the ollowing result is not entirely routine. Proposition 13 ( (Nishizawa and Power, 2007))

9 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 9 An ordinary Lawvere theory is a Lawvere Set-theory and conversely. Proo Let L be an ordinary Lawvere theory. It corresponds to a initary monad T on Set, and L is isomorphic to the restriction o Kl(T ) op to the natural numbers, with the unctor I : Nat op L given by the restriction o the canonical unctor Set Kl(T ). So I: Nat op L strictly preserves all inite limits o Nat as the corresponding inite colimits are preserved both by the inclusion into Set and by the canonical unctor into Kl(T ). So every ordinary Lawvere theory is a Lawvere Set-theory. The converse is easier: L has precisely the objects o A op, with I strictly preserving all inite limits; so L has all inite products, and they are preserved by I, although L need not have pullbacks or example. Proposition 14 The deinitions o a model o an ordinary Lawvere theory and o a Lawvere Settheory agree. Proo is both the ree category with inite products on 1 and the ree category with inite limits on 1 (Adámek and Rosický, 1994; Kelly, 1982a). So all inite product Set op preserving unctors out o Set op routinely rom Deinition 12. preserve all inite limits. The result now ollows A deinition o model o a Lawvere A-theory appeared in (Nishizawa and Power, 2007), but it was quite complex, not lowing directly rom the deinition o Lawvere A-theory. We now show that our deinition agrees with it up to coherent isomorphism. Proposition 15 Given a Lawvere A-theory I : A op L, the category Mod(L) is given, up to coherent equivalence, by the pullback in the category Cat o locally small categories. P P L [L, Set] U L A [I, Set] [A op ι, Set] Proo By Theorem 10, since A is locally initely presentable, it is equivalent to F L(A op, Set) coherently with respect to the inclusion i. Moreover, the unctor [I, Set] admits the liting o isomorphisms, i.e., or any unctor M : L Set together with a natural isomorphism o the orm MI = M : A op Set, the domain o M and the natural isomorphism extend rom A op to L. So, to give an object o the pullback P is equivalent to giving a unctor M : L Set such that the composite o M with I : A op L lies in A F L(A op, Set), i..e., such that MI preserves inite limits. But that is equivalent to giving a model o L. And that extends routinely to maps o models.

10 10 Stephen Lack and John Power By Proposition 15, up to coherent isomorphism, a model o L consists o an object X o A together with data and axioms arising rom those maps in L that are not already in A op. In practice, one usually uses this characterisation o the deinition o model, but or abstract theory, the deinition as stated here, i.e., Deinition 12, is typically more helpul. Example 16 Let A = P oset. So A is (up to equivalence) the ull subcategory o P oset determined by the inite posets. Let 0 denote the empty poset, 1 denote the one element poset, and 2 denote Sierpinski space. Consider the Lawvere P oset-theory L reely generated by two maps subject to commutativity o the ollowing two diagrams: id where the horizontal maps are the generating maps o L, the let-hand vertical map is the map in P oset op determined by the unique map in P oset rom 0 to 1 and the other two vertical maps are determined by the two maps in P oset rom 1 to 2 that choose the irst and second element o 2, labelled 0 and 1 respectively. By Proposition 15, a model o any Lawvere P oset-theory L consists o a poset P and a unctor M : L Set such that the composite unctor MI : P oset op Set is P oset(ι, P ). So M must send 0 to the one element set 1, it must send 1 to the set o elements o P, and it must send 2 to the carrier o the poset P o pairs (x, y) o elements o P or which x y, ordered pointwise. So, in particular, a model o L consists o a poset P with maps o posets 1 P P P 1 subject to commutativity o the ollowing two diagrams: P P P P t π 0 id π 1 1 P where the horizontal maps are determined by the generating maps o L and the other maps are determined by the structure o the category P oset. The commutativity o the two diagrams imply that the map P P is ully determined by the other data: it must send x to the pair (, x), where is P

11 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 11 the image o the map 1 P. So, or every element x o P, the two commutativities imply that x. Thus a model o L consists o a poset P with a least element. It will ollow that L generates the monad T on P oset or partiality. Example 16 is, by construction, an example o a Lawvere P oset-theory. The various examples o ordinary Lawvere theories o Section 2 systematically extend to become Lawvere P oset-theories too. One can see that by considering the monads on P oset generated by the various examples, then using the equivalence between initary Lawvere P oset-theories and initary monads on P oset we shall soon describe, or directly by observing that every inite set is a inite poset, and so one can regard the generating operations and equations o each example o an ordinary Lawvere theory as generating operations and equations o a Lawvere P oset-theory. Example 17 Let A = [Inj, Set]. Then A is a locally initely presentable category (see (Kelly, 1982)), used as the base category or modelling local state by O Hearn and Tennent (O Hearn and Tennent, 1997), then by Plotkin and Power (Plotkin and Power, 2002; Power, 2006). The study o state inherently involves countability as one s set V o values is typically countable. So in modelling local state, we need to generalise rom local inite presentability to local countable presentability. But A, being locally initely presentable, is necessarily locally countably presentable, and our general analysis extends routinely to countability. The category A c, which we deine to be a skeleton o the ull subcategory o A given by the countably presentable objects o A, is given, or the case o A = [Inj, Set], by the closure under countable colimits o the ull subcategory o [Inj, Set] given by the representable unctors. That may be calculated to be the ull subcategory o [Inj, Set] given by [Inj, Set c ], i.e., those unctors rom Inj to Set whose values are countable sets. In particular, or any countable set V, the constant unctor at V, which, by mild overloading o notation, we denote by V, is countably presentable, so lies in A c. The unctor L = Inj(1, ) : Inj Set, being representable, also lies in A c, as does the product L V. It ollows rom the general theory o (Kelly, 1982) that the deinition in (Plotkin and Power, 2002) o a category denoted in that paper as LS([I, Set]) systematically yields a presentation o a Lawvere [Inj, Set]-theory L LS such that the category Mod(L LS ) is the category o algebras or the monad T LS or local state on [Inj, Set]. It is not clear yet how best one can describe a denotational semantics or local state: a monad or local state has existed or some time; one o the main motivations o (Plotkin and Power, 2002) was the observation that, i one starts with operations and equations, local state can be seen semantically to extend global state; in (Power, 2006), that was taken urther by the introduction o a notion o indexed Lawvere theory; and the work in this paper suggests a still urther perspective, dispensing with the double enrichment o (Plotkin and Power, 2002) and without the explicit indexing o (Power, 2006).

12 12 Stephen Lack and John Power We have not yet developed an account o the various ways to combine Lawvere A-theories, but the sum o theories certainly exists as Law A is cocomplete, allowing the theory o (Hyland et al., 2006) to extend routinely (see (Lüth and Ghani, 2002) or an explanation o the value o the sum in unctional programming). Extending the tensor, analysed in (Hyland et al., 2006), and the distributive tensor, analysed in (Hyland and Power, 2006), will be more complex. 5 Preservation o Finite Limits There is a delicate relationship between Deinition 12 and the notions o existence and preservation o inite limits. Suppose C is a small category with inite limits, D is a small category, and H : C D preserves inite limits, but with D not necessarily having all inite limits. One can speak o the ree completion F H (D) o D under inite limits that respects the inite limits o C (Kelly, 1982a). By deinition, the category F H (D) has inite limits, and there is a canonical unctor J : D F H (D) or which the composite JH : C F H (D) preserves inite limits, and it is the universal such construct, i.e, or any small category E with inite limits and any unctor K : D E or which the composite KH preserves inite limits, there is a inite limit preserving unctor Q : F H (D) E making the triangle D J FH (D) K commute, unique up to coherent isomorphism. It ollows rom Deinition 12 that or any Lawvere A-theory I : A op L, one can characterise Mod(L) by observing that composition with J : L F I (L) yields an equivalence o categories between Mod(L) and F L(F I (L), Set)). But FL is one side o Gabriel-Ulmer duality, and with a little eort, we can extend the above observation into a relationship between the category Law A o all Lawvere A-theories and the 2-category FL, then use Gabriel-Ulmer duality to explain the relationship between models o Lawvere A-theories and initary monads on A. The details require 2-categorical care. Objects o a category are oten isomorphic to each other without being equal to each other, and so unctors between categories are oten naturally isomorphic to each other without being equal to each other. So when one considers a 2-category such as FL, one usually needs systematically to relax equalities to become isomorphisms, isomorphisms to become equivalences, unctors to become pseudo-unctors, etcetera: see (Power, 1995) or urther discussion o this in a computational setting. In particular, a pseudo-slice 2-category is the natural generalisation o the notion o slice category. Speciically, given a small category C with inite limits, the pseudo- Q E

13 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 13 slice 2-category C /FL has objects given by inite limit preserving unctors with domain C, arrows given by triangles o the orm D C = consisting o a inite limit preserving unctor rom D to D and a natural isomorphism between the two unctors rom C to D : in particular, note that the diagram need not commute, and the isomorphism inside it is part o the data. The 2-cells o the pseudo-slice 2-category C /FL are given by natural transormations that respect the isomorphisms in the respective triangles. Using that deinition, and systematically relaxing the notion o unctor between categories to pseudo-unctor between 2-categories, similarly or natural transormations, we have the ollowing. Theorem 18 The category Law A is a pseudo-corelective subcategory o the pseudo-slice 2- category A op /FL Law A A op /FL where a Lawvere A-theory I : A op D under inite limits that respects the inite limits o A op sends a inite limit preserving unctor A op aithul)-actorisation. L is sent to the ree completion F I(L) o L, and the pseudo-corelection C to its (identity-on-objects, ully In general, it is not easy to give a concrete characterisation o the ree completion F I (L) o a Lawvere A-theory under inite limits. But we can give a general description o F I (L), albeit not a concrete one. It ollows rom Proposition 15 that, or any Lawvere A-theory L, the Yoneda embedding restricts to a ully aithul unctor Y : L op Mod(L) It then ollows rom Gabriel-Ulmer duality that F I (L) op is the ull subcategory o Mod(L) given by closing L op under inite colimits in Mod(L). Colimits in Mod(L) are generally awkward to calculate, although iltered colimits are easy (see Theorem 20). But we do not need a concrete description o F I (L) anyway; we only ever use its deining universal property. As an indication o how the paper is to develop rom here, compare Theorem 18 with the ollowing. Theorem 19 The category Mnd (A) o initary monads on a locally initely presentable category A is a pseudo-corelective subcategory o (LocPres /A) op Mnd (A) (LocPres /A) op where a initary monad T is sent to the orgetul unctor T -Alg A and a iltered

14 14 Stephen Lack and John Power colimit preserving unctor G : B A with let adjoint F is sent to the monad GF. 6 Theories and Monads We now work towards relating Lawvere A-theories with monads on A, extending the main result o (Nishizawa and Power, 2007) by explaining it in the light o Theorems 18 and 19. Theorem 20 For any Lawvere A-theory I : A op L, the category Mod(L) is locally initely presentable and the unctor U L : Mod(L) A is a map o locally initely presentable categories. Proo Let F I (L) denote the ree completion o L under inite limits that respects the inite limits o A op, c Theorem 18. It ollows immediately rom the universal property o F I (L) that Mod(L) is equivalent to F L(F I (L), Set), which, by Theorem 8, is locally initely presentable. Moreover, U L is determined by composition with the canonical composite unctor A op L F I(L) which preserves inite limits by construction. So, by a urther application o Theorem 8, U L is a map o locally initely presentable categories. Theorem 20 is undamental, yielding a string o corollaries. The proo implies a little more than the theorem as stated. Speciically, it yields the ollowing. Corollary 21 For any Lawvere A-theory I : A op L, the two vertical unctors U L and [I, Set] in the diagram o Proposition 15 have let adjoints, yielding a square that is commutative up to natural isomorphism as ollows: Mod(L) PL [L, Set] F L A = Lan I [A op ι, Set] where Lan I denotes the let Kan extension along the unctor I (Kelly, 1982). Corollary 22 For any Lawvere A-theory I : A op L, the unctor U L : Mod(L) A is initarily monadic. Proo

15 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 15 By Theorem 20, the unctor U L is initary and has a let adjoint. Let, g be a U L -split coequaliser pair in Mod(L). Since [L, Set] is cocomplete, P L and P L g have a coequaliser, and the coequaliser can be chosen so that it is strictly preserved by [I, Set]. Since a split coequaliser o U L and U L g is also preserved by ι, and g have a coequaliser in Mod(L) and U L strictly preserves it. So by Beck s monadicity theorem (Barr and Wells, 1985), U L is monadic. Let T L be the initary monad on A induced by L. The construction o T L rom L extends routinely to a unctor T : Law A Mnd (A) We can combine that unctor with the pseudo-unctors in Theorems 8, 18 and 19 as ollows. Corollary 23 The diagram Law A T Mnd (A) commutes. F I ( ) A op /FL ( )-Alg (LocPres /A) op Gabriel-Ulmer duality, Theorem 8, asserts that the bottom line o the diagram is a biequivalence o 2-categories. The central theorem o (Nishizawa and Power, 2007) asserts that the top line is an equivalence o categories. The main line o results goes as ollows. Corollary 24 For any Lawvere A-theory I : A op L, one recovers Iop : A L op rom F L as the (identity-on-objects, ully aithul) actorisation o F L ι. L op I op A I Mod(L) ι F L A Proo For any inite limit preserving unctor H : C D, Gabriel-Ulmer duality, Theorem 8, asserts that H is the restriction o F : F L(C, Set) F L(D, Set), where F is the let adjoint to the unctor F L(H, Set) : F L(D, Set) F L(C, Set) given by composition with H. Considering the special case where C = A op, D = F I (L), and H is the canonical composite, it ollows rom Corollary 21 that the

16 16 Stephen Lack and John Power diagram commutes i I is taken to be the Yoneda embedding regarded as having codomain in Mod(L). Fully aithulness o I ollows rom ully aithulness o the Yoneda embedding. So or an arbitrary initary monad T on A, deine (K T, I T, ι T ) by taking the (identity-on-objects, ully aithul) actorisation o F T ι: ι T K T Kl(T ) I T T F A A ι Since ι and F T preserve inite colimits and ι T relects inite colimits, I T is an identity-on-objects strict inite colimit preserving unctor. So we deine L T to be K op T. Theorem 25 ( (Nishizawa and Power, 2007)) For a initary monad T on A, let F T G T be the canonical adjunction between the Eilenberg-Moore category T -Alg and A, and let Q T send a T -algebra α to T -Alg(ι T, α). Then, i we allow Q T to be replaced by a canonically isomorphic unctor, the ollowing square yields a pullback: T -Alg Q T [L T, Set] G T A [I op T ι, Set] [A op, Set] Corollary 26 The construction o T L rom an arbitrary Lawvere A-theory L and that o L rom an arbitrary initary monad T on A extend canonically to an equivalence o categories Law A Mnd (A). Moreover, the categories Mod(L) and T L -Alg are coherently equivalent. Proo By Theorem 25, T = T LT or an arbitrary initary monad T on A. Conversely, given an arbitrary Lawvere A-theory L, the Lawvere A-theory L TL is deined to be the (identity-on-objects, ully aithul) actorisation o F T L ι: A T L -Alg. By Corollary 24 and since Mod(L) T L -Alg, this actorisation agrees with L, and so L TL is isomorphic to L. The two constructions routinely extend to an equivalence o categories. 7 Change o Base In this section, urther developing our axiomatisation and extension o the relationship between ordinary Lawvere theories and monads on Set, we consider the eect o change o base. Speciically, given locally initely presentable categories A and B and a map o locally initely presentable categories rom A to B, i.e., a iltered

17 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 17 colimit preserving unctor U : A B that has a let adjoint F, we study the relationship between Lawvere A-theories, equivalently initary monads on A, and Lawvere B-theories, equivalently initary monads on B, as induced by U. Every map F U : A B o locally initely presentable categories routinely induces a 2-unctor (LocPres /A) op (LocPres /U) op (LocPres /B) op This 2-unctor has a let biadjoint, which we denote by U, given by pseudopullback, which, in the situation o primary interest to us, is equivalent to an ordinary pullback (Joyal and Street, 1993). Trivially, every initary monad T A on A induces a initary monad UT A F on B. So there must be a corresponding construct or Lawvere theories cohering with the inclusion pseudo-unctors o Theorems 18 and 19. The coherence is subtle. In terms o monads, it is as ollows. Theorem 27 Every map F U : A B o locally initely presentable categories canonically induces an adjunction or which the diagram o let adjoints F mnd U mnd : Mnd (A) Mnd (B) Mnd (B) F mnd Mnd (A) commutes. (LocPres /B) op U (LocPres /A) op Proo U mnd sends a initary monad T A on A to UT A F with the evident monad structure on it. And F mnd sends a initary monad T B on B to the monad induced by considering the pullback P T -Alg A U B in Cat and observing that P is initarily monadic over A, then taking the induced monad. Commutativity o the diagram ollows by construction o F mnd, c (Joyal and Street, 1993). Commutativity o the diagram in Theorem 27, subject to a mild 2-categorical

18 18 Stephen Lack and John Power subtlety (Joyal and Street, 1993), determines a canonical 2-natural transormation Mnd (A) U mnd Mnd (B) (LocPres /A) op (LocPres /U) op (LocPres /B) op The component at T A o that 2-natural transormation is the canonical comparison unctor T A -Alg UT A F -Alg The above relationships can duly be expressed equally in terms o Lawvere A- theories and Lawvere B-theories as ollows. Theorem 28 Every map F U : A B o locally initely presentable categories canonically induces an adjunction or which the diagram o let adjoints F Law U Law : Law A Law B Law B F Law Law A commutes. (LocPres /B) op U (LocPres /A) op Proo U Law sends a Lawvere A-theory I : A op L A to the (identity-on-objects, ully aithul)-actorisation o the composite B op A op L A And F Law sends a Lawvere B-theory I : B op L B to the Lawvere A-theory generated by taking the pushout B op F A op I L B J P and then taking F J (P ). Commutativity o the diagram in the statement o the theorem ollows by construction o F Law.

19 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 19 Commutativity o the diagram in Theorem 28 determines a canonical 2-natural transormation Law A U Law LawB (LocPres /A) op (LocPres (LocPres /U) op /B) op The component at L A o that natural transormation is determined by composition. Example 29 The orgetul unctor U : P oset Set is a map o locally initely presentable categories. Its let adjoint F takes a set X to itsel regarded as a discrete poset. So the unctor U mnd sends a monad T on P oset to the monad on Set that sends a set X to the underlying set o T X. For instance, it sends the monad (S ) S on P oset or global state to the monad on Set or global state. The behaviour o F mnd seems less natural in regard to computational eects as it is determined by its behaviour on T -Alg rather than on Kl(T ): the monad F mnd necessarily exists, but we do not have any comprehensible concrete description o it in general: given a monad T on Set, the monad F mnd (T ) on P oset sends a poset P to the ree poset Q equipped with T -structure on the underlying set UQ o Q. Example 30 A class o examples o change o base arises when one considers local state. In this paper, ollowing (Plotkin and Power, 2002; Power, 2006), we have ocused on [Inj, Set] as an appropriate base category in which to study local state. But it is not the only base preshea category to have been used. For instance, the categories [Nat, Set] and [Iso, Set], where Iso is the category o natural numbers and permutations, or more complex variants, primarily in the work o O Hearn and Tennent, have appeared (O Hearn and Tennent, 1997). Change o base applies to these. Any unctor H : C D between small categories C and D generates a map [H, Set] : [D, Set] [C, Set] o locally initely presentable categories, with the let adjoint to [H, Set] given by let Kan extension. So, applying Theorems 27 and 28, H induces adjunctions and F mnd [H, Set] mnd : Mnd ([D, Set]) Mnd ([C, Set]) F Law [H, Set] Law : Law [D,Set] Law [C,Set] In particular, or example, H might be the inclusion o Inj into Nat, thus yielding an adjunction between Mnd ([D, Set]) and Mnd ([C, Set]), equivalently between Law [D,Set] and Law [C,Set]. There is a second, more delicate approach to change o base as ollows. We have

20 20 Stephen Lack and John Power analysed the category Mod(L) or a Lawvere A-theory L and shown that it supports a orgetul unctor U L : Mod(L) A. We have urther shown that U L is initarily monadic, and the construction o U L characterises the initary monads on A. But in Section 2, we considered models o an ordinary Lawvere theory in any base category with inite products, not only in Set. So one wonders whether we can consider models o a Lawvere A-theory in categories other than A. In act, we can do that, but the situation is subtle. Let A be a locally initely presentable category and let I : A op category o the orm F L(A op category Mod(L, F L(A op L be a Lawvere A-theory. Consider any, B) where B has inite limits. We can deine the, B)) o models o L in F L(Aop, B) to be the pullback Mod(L, F L(A op, B)) PL [L, B] U L F L(A op, B) [I, B] inc [A op, B] generalising the characterisation in Proposition 15 o the category o models o L in A. A priori, this may look special, not recovering the idea o a model o an ordinary Lawvere theory in an arbitrary category with inite products as in Deinition 1. But that is illusory: i A = Set, the category A op is Nat op, which is the ree category with inite limits on 1. So, or any category B with inite limits, F L(A op, B) is equivalent to B. And so, in the case o A = Set, the generality we assert here means we can take models o a Lawvere A-theory in any category B with inite limits. Thus the generality we propose here covers all examples o interest to us. With care, we can go even urther: both in Deinition 1 and here, we do not actually need all inite products or all inite limits in B respectively: we just need some speciic ones. So, with care, it is routine give a urther generalisation beyond the assertion that B has inite limits to include Deinition 1 entirely, but the lack o examples makes it seem complex to the point o distraction to give the details here. 8 Conclusions The notion o Lawvere theory, as introduced by Lawvere in his Ph.D. thesis (Lawvere, 1963), has become increasingly valuable over recent years in analysing computational eects, allowing a more reined denotational semantics than that provided by monads (Hyland et al., 2007; Hyland et al., 2006). Classically, the relationship between Lawvere theories and monads has only been properly understood or base category Set, more recently or base V -category V (Power, 2000). That does not ully cover the range o situations in which one seeks to model eects, as, in particular, local eects are typically modelled in preshea categories such as [Inj, Set],

21 Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base 21 with enrichment in [Inj, Set] looking out o place (O Hearn and Tennent, 1997; Plotkin and Power, 2002; Power, 2006). So, in this paper, extending mathematical ideas rom (Nishizawa and Power, 2007), we have addressed the situation, developing a notion o Lawvere A-theory, where one does not insist upon enrichment o the base category A in itsel. To give a mathematically uniied account o the situation led us to explicate Gabriel- Ulmer duality, as it yields an account o change o base by considering pseudo-slice 2-categories. This is one o a number o recent extensions o the notion o Lawvere theory, others being given by discrete Lawvere theories (Hyland and Power, 2006) and indexed Lawvere theories (Power, 2006). Each o these extensions has been devised with particular applications in mind, all o them relevant to computational eects. It is not clear yet precisely what combined extension o the notion o Lawvere theory might be optimal. So that remains an open question, partly because the various mathematical developments have given rise to new computational questions, such as the classiication o eects into constructors, deconstructors, and logical eects mooted in (Hyland et al., 2006). Reerences Adámek, J. and Rosický, J. (1994) Locally Presentable and Accessible Categories. London Mathematical Society Lecture Note Series, Vol. 189, Cambridge University Press. Barr, M. and Wells, C. (1985) Toposes, Triples and Theories, Springer-Verlag. Barr, M. and Wells, C. (1990) Category Theory or Computing Science, Prentice-Hall. Benton, N., Hughes, J. and Moggi, E. (2000) Monads and Eects, In Proc APPSEM 2000, Lecture Notes in Computer Science 2395, pp , Springer-Verlag. Heckmann, R.(1994) Probabilistic Domains, In Proc. CAAP 94, Lecture Notes in Computer Science 136, pp , Springer-Verlag. Hyland, M., Levy, P. B., Plotkin, G. D. and Power, A. J. (2007) Combining Continuations with other eects, John Reynolds Festschrit. Hyland, J. M. E., Plotkin, G. D., and Power, A. J. (2006) Combining Computational Eects: Sum and Tensor, Theoretical Computer Science 357: pp Hyland, J. M. E and Power, A. J. (2006) Discrete Lawvere Theories and Computational Eects, Theoretical Computer Science 366: pp Hyland, J. M. E and Power, A. J. (2007) The Category-Theoretic Understanding o Universal Algebra: Lawvere Theories and Monads, Electronic Notes in Theoretical Computer Science 172, pp Joyal, A. and Street, R. (1993) Pullbacks Equivalent to Pseudo-Pullbacks, Cahiers de Topologie et Géométrie Diérentielle Catégoriques 34: pp Kelly, G. M. (1982) Basic Concepts o Enriched Category Theory, Cambridge University Press. Kelly, G. M. (1982a) Structures Deined by Finite Limits in the Enriched Context I, Cahiers de Topologie et Géométrie Diérentielle 23 (1): pp Lawvere, F.W. (1963) Functorial Semantics o Algebraic Theories, Proc. Nat. Acad. Sciences USA 50 (5): pp Lüth, C. and Ghani, N. (2002) Monads and Modularity, In Proc. FroCos 2002, Lecture Notes in Artiicial Intelligence 2309, pp , Springer-Verlag.

22 22 Stephen Lack and John Power Moggi, E. (1989) Computational Lambda-Calculus and Monads, In Proc. LICS 89, pp , IEEE Press. Moggi, E. (1991) Notions o Computation and Monads, In. and Comp. 93 (1): pp Nishizawa, K. and Power, A. J. (2007) Lawvere Theories Enriched over a General Base, J. Pure Appl. Algebra (to appear). O Hearn, P. W., and Tennent, R. D. (1997) Algol-like Languages, Birkhauser. Plotkin, G. D. and Power, A. J. (2002) Notions o Computation Determine Monads, In Proc. FOSSACS 02, Lecture Notes in Computer Science 2303, pp , Springer- Verlag. Plotkin,G. D. and Power, A. J. (2003) Algebraic Operations and Generic Eects, Applied Categorical Structures 11: pp Power, A. J. (1995) Why Tricategories? Inormation and Computation, 120: pp Power, A. J. (2000) Enriched Lawvere Theories, Theory and Applications o Categories, 6: pp Power, A. J. (2006) Semantics or local computational eects, Proc. MFPS 2006, Electronic Notes in Theoretical Computer Science 158, pp Robinson, E. (2002) Variations on Algebra: Monadicity and Generalisations o Equational Theories, In Formal Aspects o Computing 13 (3,4,5), pp