A Logic of Injectivity

Transcription

1 A Logic of Injectivity J. Adámek, M. Hébert and L. Sousa Abstract Injectivity of objects wit respect to a set H of morpisms is an important concept of algebra, model teory and omotopy teory. Here we study te logic of injectivity consequences of H, by wic we understand morpisms suc tat injectivity wit respect to H implies injectivity wit respect to. We formulate tree simple deduction rules for te injectivity logic and for its finitary version were morpisms between finitely ranked objects are considered only, and prove tat tey are sound in all categories, and complete in all reasonable categories. 1 Introduction Recall tat an object A is injective w.r.t. a morpism : P P provided tat every morpism from P to A factors troug. We address te following problem: given a set H of morpisms, wic morpisms are injectivity consequences of H in te sense tat every object injective w.r.t. all members of H is also injective w.r.t.? We denote te injectivity consequence relationsip by H =. Tis is a classical topic in general algebra: te equational logic of Garrett Birkoff [10] is a special case. In fact, an equation s = t is a pair of elements of a free algebra F, and tat pair generates a congruence on F. An algebra A satisfies s = t iff it is injective w.r.t. te canonical epimorpism : F F/. Tus, if we restrict our sets H to regular epimorpisms wit free domains, ten te logic of injectivity becomes precisely te equational logic. However, tere are oter important cases in algebra: recall for example te concept of injective module, were H is te set of all monomorpisms (in te category of modules). To mention an example from omotopy teory, recall tat a Kan complex [14] is a simplicial set injective w.r.t. all te monomorpisms k n n (for n, k N, k n) were n is te complex generated by a single n-simplex and k n is te subcomplex obtained by deleting te k-t 1-simplex and all adjacent faces. We can ask for example weter Kan complexes can be specified by a simpler collection of monomorpisms, as a special case of our injectivity logic. Te tird autor acknowledges financial support by te Center of Matematics of te University of Coimbra and te Scool of Tecnology of Viseu 1

2 Injectivity establises a Galois correspondence between objects and morpisms of a category. Te closed families on te side of objects are called injectivity classes: for every set H of morpisms we obtain te injectivity class InjH, i.e., te class of all objects injective w.r.t. H. In [5] small-injectivity classes in locally presentable categories were caracterized as precisely te full accessible subcategories closed under products, and in [18] tis was sarpened in te following sense. Let us call a morpism λ-ary if its domain and codomain are λ-presentable objects. Injectivity classes wit respect to λ-ary morpisms are precisely te full subcategories closed under products, λ-filtered colimits, and λ-pure subobjects. For injectivity w.r.t. cones or trees of morpisms similar results are in [7] and [15]. In te present paper we study closed sets on te side of morpisms, i.e., we develop a deduction system for te above injectivity consequence relationsip =. It as altogeter tree deduction rules, wic are quite intuitive. Firstly, observe tat every object injective w.r.t. a composite = 2 1 is injective w.r.t. te first morpism 1. Tis gives us te first deduction rule cancellation It is also easy to see tat injectivity w.r.t. implies injectivity w.r.t. any morpism opposite to in a pusout (along an arbitrary morpism), wic yields te rule pusout for every pusout Finally, an object injective w.r.t. two composable morpisms is also injective w.r.t. teir composite. Te same olds for tree, four,... morpisms but also for a transfinite composite as used in omotopy teory. For example, given an ω-cain of morpisms A 0 0 A 1 1 A ten teir ω-composite is te first morpism c 0 : A 0 C of (any) colimit cocone c n : A n C (n N) of te cain. Observe tat c 0 is indeed an injectivity consequence of { i ; i < ω}. For every ordinal λ we ave te concept of a λ-composite of morpisms (see 2.10 below) and te following deduction rule, expressing te fact tat an object injective w.r.t. eac i is injective w.r.t. te transfinite composite: transfinite composition i (i < λ) for every λ-composite of ( i ) i<λ We are going to prove tat te Injectivity Logic based on te above tree rules is sound and complete. Tat is, given a set H of morpisms, ten H = olds for precisely tose morpisms wic can be proved from assumptions in H using te tree deduction rules above. Tis olds in a number of categories, e.g., in (a) every variety of algebras, (b) te category of topological spaces and many nice subcategories (e.g. Hausdorff spaces), and 2

3 (c) every locally presentable category of Gabriel and Ulmer. We introduce te concept of a strongly locally ranked category encompassing (a)-(c) above, and prove te soundness and completeness of our Injectivity Logic in all suc categories. Observe tat te above logic is infinitary, in fact, it as a proper class of deduction rules: one for every ordinal λ in te instance of transfinite composition. We also study, following te footsteps of Grigore Roşu, te completeness of te corresponding Finitary Injectivity Logic: it is te restriction of te above logic to λ finite. Well, all we need to consider are te cases λ = 2, called composition, and λ = 0, called identity: composition 1 0 for = 1 0 identity id A Te resulting finitary deductive system (introduced in [6] as a sligt modification of te deduction system of Grigore Roşu [19]) as four deduction rules; it is clearly sound, and te main result of our paper (Teorem 6.2) says tat it is also complete wit respect to finitary morpisms, i.e., morpisms wit domain and codomain of finite rank. Tis implies te expected compactness teorem: every finitary injectivity consequence of a set H of finitary morpisms is an injectivity consequence of some finite subset of H. Te completeness teorem for Finitary Injectivity Logic will ten be extended to te k-ary Injectivity Logic, defined in te expected way. Ten te full completeness teorem easily follows. Te fact tat te full Injectivity Logic above is complete in strongly locally ranked categories can also be derived from Quillen s Small Object Argument [17], see Remark 3.9 below. However our sarpening to te k-ary logic for every cardinal k cannot be derived from tat paper, and we consider tis to be a major step. Related work Bernard Banascewski and Horst Herrlic sowed tirty years ago tat implications in general algebra can be expressed categorically via injectivity w.r.t. regular epimorpisms, see [9]. A generalization to injectivity w.r.t. cones or even trees of morpisms was studied by Hajnal Andréka, István Németi and Ildikó Sain, see e.g. [7, 8, 15]. To see more precisely ow tat work relates to ours and to classical logic, consider injectivity in te category of all Σ-structures (and Σ-omomorpisms), were Σ is any signature. Ten recall from [4], 5.33 tat tere is a natural way to associate to a (finitary) morpism f : A B a (finitary) sentence f := X( A (X) Y ( B (X, Y ))) (were A (X) and B (X, Y ) are sets of atomic formulas) suc tat an object C satisfies f if and only if it is injective wit respect to f (see 2.22 below for more on tis). Suc sentences are called regular sentences. In tis paper we concentrate on te proof teory for te (finite and infinite) regular logics. As mentioned above, te restriction 3

4 to epimorpisms correspond to considering only te quasi-equations (i.e., no existential quantifiers), and just equations if we impose tey ave projective domains. Recently, Grigore Roşu introduced a deduction system for injectivity, see [19], and e proved tat te resulting logic is sound and complete for epimorpisms wic are finitely presentable, see 3.5, and ave projective domains. A sligt modification of Roşu s system was introduced in [6]: tis is te deduction system 2.4 below. It differs from [19] by formulating pusout more generally and using composition in place of Roşu s union. In [6] completeness is proved for sets of epimorpisms wit finitely presentable domains and codomains. (Tis is sligtly stronger tan requiring te epimorpisms to be finitely presentable, owever, witout te too restrictive assumption of projectivity of te domains te logic fails to be complete for finitely presentable epimorpisms in general, see [6].) In te present paper completeness of te finitary logic is proved for arbitrary morpisms (not necessarily epimorpisms) wit finitely presentable domains and codomains. Te fact tat te assumption of epimorpism is dropped makes te proof substantially more difficult. We present a sort proof in locally presentable categories first, and ten a proof of a more general result for strongly locally ranked categories. We also formulate te appropriate infinitary logic dealing wit arbitrary morpisms. Tere are oter generalizations of Birkoff s equational logic wic are, except for te common motivation, not related to our approac. For example te categorical approac to logic of (ordered) many-sorted algebras of Razvan Diaconescu [11], and te logic of implications in general algebra of Robert Quackenbus [16]. In our joint paper [1] we are taking anoter route to generalize te equational logic: we consider ortogonality of objects to a morpism instead of injectivity. Te deduction system is similar: te rule cancellation as to be weakened, and an additional rule concerning coequalizers is added. We prove te completeness of te resulting logic of ortogonality in locally presentable categories. Te corresponding sentences are te so called limit sentences, X( A (X)!Y ( B (X, Y ))), were!y means tere exists exactly one Y suc tat. 2 Logic of injectivity 2.0. Assumption Trougout te paper we assume tat we are working in a cocomplete category Definition A morpism is called an injectivity consequence of a set of morpisms H, notation H = provided tat every object injective w.r.t. all morpisms in H is also injective w.r.t Examples (1) A composite = 2 1 is an injectivity consequence of { 1, 2 }. (2) Conversely, in every composite = 2 1 te morpism 1 is an injectivity 4

5 consequence of : (3) In every pusout A 1 A 2 A X A A u B v B is an injectivity consequence of : A A u v B B X 2.3. Remark Te above examples are exaustive. More precisely, te following deduction system, introduced in [6], see also [19], (were, owever, it was only applied to epimorpisms) will be proved complete below: 2.4. Definition Te Finitary Injectivity Deduction System consists of one axiom identity id A and tree deduction rules composition cancellation if is defined and pusout if We say tat a morpism is a formal consequence of a set H of morpisms (notation H ) in te Finitary Injectivity Logic if tere exists a proof of from H (wic means a finite sequence 1,..., n = of morpisms suc tat for every i = 1,..., n te morpism i lies in H or is a conclusion of one of te deduction rules wose premises lie in { 1,..., i 1 }). 5

6 2.5. Lemma Te Finitary Injectivity Logic is sound, i.e., if a morpism is a formal consequence of a set of morpisms H, ten is an injectivity consequence of H. Briefly: H implies H =. Te proof follows from Remark Later we define finitary morpisms (as morpisms wose domains and codomains are finitely presentable (Section 3) or of finite rank (Section 5)), and in Section 6 we prove tat te resulting Finitary Injectivity Logic is complete, i.e., tat H = implies H for every set H of finitary morpisms and every finitary Example Te following rule finite coproduct (were for i : A i B i te morpism : A 1 + A 2 B 1 + B 2 is te canonical coproduct morpism) is obviously sound. Here is a proof in te Finitary Injectivity Logic: Using te pusouts 1 A 1 B 1 A 1 + A 2 1 +id A2 B 1 + A 2 2 A 2 B 2 B 1 + A 2 id B1 + 2 B 1 + B 2 we can write id A2 id B via pusout via composition since = (id B1 + 2 ) ( 1 + id A2 ) Example Te following rule finite wide pusout for every wide pusout 1... n 1 2 k 1 k 2 C... n... k n were = k i i is sound. Here is a proof in te Finitary Injectivity Logic: If n = 2 we ave 6

7 1 2 k 2 = k 2 2 via pusout via composition If n = 3 denote by r a pusout of 1, 2, ten a pusout, 3, 1 r 3 2 k 3 k 1 k 2 3 of 3 along r forms a wide pusout of 1, 2 and 3 : k 2 r k 3 = k 3 3 via pusout via composition via pusout via composition Etc Remark We want to define a composition of a cain of λ morpisms for every ordinal λ (see te case λ = ω in te Introduction). Recall tat a λ-cain is a functor A from λ, te well-ordered category of all ordinals i < λ. Recall furter tat λ + denotes te successor ordinal, i.e., te set of all i λ Definition (i) We call a λ-cain A smoot if for every limit ordinal i < λ we ave A i = colim j<i A j wit te colimit cocone of all a ji = A(j i). (ii) A morpism is called a λ-composite of morpisms ( i ) i<λ, were λ is an ordinal, if tere exists a smoot λ + -cain A wit connecting morpisms a ij : A i A j for i j λ suc tat i = a i,i+1 for all i < λ and = a 0,λ Examples λ = 0: No morpism i is given, just an object A 0 ; and = a 0,0 is te identity morpism of A 0. λ = 1: A morpism 0 is given, and we ave = a 0,1 = 0. Tus, a 1-composite of 0 is 0. 7

8 λ = 2: Tis is te usual concept of composition: given morpisms 0, 1, teir 2-composite exists iff tey are composable. Ten 1 0 is te 2-composite. λ = ω: Tis is te case mentioned in te Introduction. Observe tat, unlike te previous cases, an ω-composite is only unique up to isomorpism Lemma A λ-composite of morpisms ( i ) i<λ is an injectivity consequence of tese morpisms. Proof Tis is a trivial transfinite induction on λ. In case λ = 0 tis states tat id A is an injectivity consequence of, etc Definition Te Injectivity Deduction System consists of te deduction rules cancellation pusout for every pusout and te rule sceme (one rule for every ordinal λ) transfinite composition i (i < λ) for every λ-composite of ( i ) i<λ We say tat a morpism is a formal consequence of a set H of morpisms (notation H ) in te Injectivity Logic if tere exists a proof of from H (wic means a cain ( i ) i n of morpisms, were n is an ordinal, suc tat = n, and eac i eiter lies in H, or is a conclusion of one of te deduction rules wose premises lie in { j } j<i ) Lemma Te Injectivity Logic is sound, i.e., if a morpism is a formal consequence of a set H of morpisms, ten is an injectivity consequence of H. Briefly: H implies H =. Te proof (using 2.12) is elementary Remark In 2.13 we can replace transfinite composition by te deduction rule wide pusout, see below, wic makes use of te (obvious) fact tat an object A injective w.r.t. a set { i } i<λ of morpisms aving a common domain is also injective w.r.t. teir wide pusout. Let us note ere tat tis rule does not replace pusout of 2.13 (because in te latter a pusout of along an arbitrary morpism is considered) Definition Te deduction rule wide pusout i (i < λ) for a wide pusout of { i } i<λ applies, for every cardinal λ, to an arbitrary object P and an arbitrary set { i } of λ 8

9 morpisms wit te common domain P and te following wide pusout i P i k i P Q... = k i i (for any i) Remark Again, tis is a sceme of deduction rules: for every cardinal λ we ave one rule λ-wide pusout. Observe tat λ = 0 yields te rule identity Lemma Te Injectivity Deduction System 2.13 is equivalent to te deduction system composition, cancellation, pusout and wide pusout. Proof (1) We can derive wide pusout from For every ordinal number λ we derive te rule i (i < λ) for a wide pusout of { i } i<λ by transfinite induction on te ordinal λ. We are given an object P and morpisms i : P P i (i < λ). Te case λ = 0 is trivial, from λ derive λ + 1 by using pusout, and for limit ordinals λ form te restricted multiple pusouts Q j of morpisms i for i < j, and observe tat tey form a smoot cain wose composite is a multiple pusout of all i s. (2) From te system in 2.17 we can derive te rule λ-composition, were λ is an arbitrary ordinal: te case λ = 0 follows from 0-wide pusout. Te isolated step uses composition: te (λ + 1)-composite of ( i ) i λ is simply λ k were k is te λ-composite of ( i ) i<λ. In te limit case, use te fact tat a composite of ( i ) i<λ is a wide pusout of {k i } i<λ, were k i is a composite of ( j ) j<i Remark For every infinite cardinal k te k-ary Injectivity Deduction System is te system 2.13 were λ ranges troug ordinals smaller tan k. A proof of a morpism from a set H in te k-ary Injectivity Logic is, ten, a proof of lengt n < k using only te deduction rules wit λ restricted as above. Te last lemma can, obviously, be formulated under tis restriction in case we use te sceme λ-wide pusout for all cardinals λ < k Definition Te deduction rule coproduct i (i < λ) i<λ i applies, for every cardinal λ, to an arbitrary collection of λ morpisms i : A i B i. 9

10 2.20. Lemma Te Injectivity Deduction System 2.13 is equivalent to te deduction system of 2.17 wit wide pusout replaced by identity + coproduct Proof (1) coproduct follows from In fact, i<λ i : i<λ A i i<λ B i is a wide pusout of te morpisms k j : i<λ A i i<j A i + B j + j<i<λ A i, were j ranges troug λ, wit components id Ai (i j) and j, and k j is a pusout of j along te j-t coproduct injection of i<λ A i. (2) Conversely, wide pusout follows from identity+coproduct. We obviously need to consider only λ > 1 and ten we use te fact tat given morpisms i : A B i (i < λ), teir wide pusout : A C can be obtained from i<λ i by pusing out along te codiagonal : λ A A: A ` i Bi A C Remark Te deduction system of te last lemma as five rules, but te advantage against te system 2.13 is tat tey are particularly simple to formulate: identity cancellation id A composition if 2 1 is defined pusout given coproduct i (i I) i I i We prove below tat 2.13, and terefore te above equivalent deduction system, is not only sound but (in a number of categories) also complete Remark To relate our deduction rules to te usual ones (of classical logic), let us consider, as in te Introduction, te category of all Σ-structures. Ten any object A can be presented by a set A (X) of atomic formulas wit parameters X in A: for te familiar algebraic structures, tis is just te usual concept of generators and relations. Given a morpism f : A B, and suc presentations A (X) and B o(y ) of A and B, we can also present B by B (X, Y ), wic is te union of B o(y ) and te set of all te 10

11 equations x = t(y ) for wic f(x) = t(y ) (t a Σ-term). Ten for te sentence f := X( A (X) Y ( B (X, Y ))) we ave tat an object C is f-injective iff C = f. Note tat if f is finitary (see te Introduction or 3.4 below), te presentations, and ence f, can be cosen to be finitary (more details in [4], 5.33). Now, we can associate Gentzen-style rules to sets of atomic formulas, generalizing te idea of wat was done (wit more accuracy) in [6] for sets of equations: associating A (X) B (X, Y ) to X( A (X) Y ( B (X, Y ))), te identity axiom is of course A (X) A (X) ; cancellation is a categorical version of te restriction rule A (X) (B (X, Y ) C (X, Y, Z)) A (X) B (X, Y ) ; pusout is essentially te weakening rule A (X) B (X, Y ) (A (X) C (X, Z)) B (X, Y ) ; and composition is a cut rule A (X) B (X, Y ), B (X, Y ) C (X, Y, Z) A (X) C (X, Y, Z). Te usual stronger cut rule corresponds to A (X) B (X, Y ), ((B (X, Y ) C (X, Y, Z)) D (X, Y, Z, U) (A (X) C (X, Y, Z)) D (X, Y, Z, U) wic is proved via A f B, B + C g D A + C g (f+1 C) C, f g f + id C g g (f + 1 C ) pusout composition 11

12 3 Completeness in locally presentable categories 3.1. Assumption In te present section we study injectivity in a locally presentable category A of Gabriel and Ulmer, see [12] or [4]. Tis means tat: (a) A is cocomplete, and (b) tere exists a regular cardinal λ suc tat A as a set of λ-presentable objects wose closure under λ-filtered colimits is all of A. Recall tat an object A is λ-presentable if its om-functor om(a, ) : A Set preserves λ-filtered colimits. Tat is, given a λ-filtered diagram D wit a colimit c i : D i C (i I) in A, ten for every morpism f : A C and (i) a factorization of f troug c i exists for some i I, (ii) factorizations are essentially unique, i.e., given i I and c i g = c i g for some g, g : A D i, tere exists a connecting morpism d ij : D i D j of te diagram wit d ij g = d ij g Examples (see [4]) Sets, preseaves, varieties of algebras and simplicial sets are examples of locally presentable categories. Categories suc as Top (topological spaces) or Haus (Hausdorff spaces) are not locally presentable Remark (a) In te present section we prove tat te Injectivity Logic is complete in every locally presentable category. Te reader may decide to skip tis section since we prove a more general result in Section 6. Bot of our proofs are based on te fact tat for every set H of morpisms te full subcategory InjH (of all objects injective w.r.t. morpisms of H) is weakly reflective. Tat is: every object A A as a morpism r : A A, called a weak reflection, suc tat and (i) A lies in InjH (ii) every morpism from A to an object of InjH factors troug r (not necessarily uniquely). In te present section we will utilize te classical Small Object Argument of D. Quillen [17]: tis tells us tat every object A as a weak reflection r : A A in InjH suc tat r is a transfinite composite of morpisms of te class Ĥ = {k; k is a pusout of a member of H along some morpism}. (b) Te reason for proving te completeness based on te Small Object Argument in te present section is tat te proof is sort and elegant. However, by using a more refined construction of weak reflection in InjH, wic we present in Section 5, we will be 12

13 able to prove te completeness in te so-called strongly locally ranked categories, wic include Top and Haus. Te spirits of te two proofs are quite different. Given an injectivity consequence of a set of morpisms, in tis section we will sow ow to derive a formal proof of from Quillen s construction of te weak reflection; tis construction is linear, forming a transfinite composite. In te next section, a weak reflection will be constructed as a colimit of a filtered diagram wic someow presents simultaneously all te possible formal proofs Definition A morpism is called λ-ary provided tat its domain and codomain are λ-presentable objects. For λ = ℵ 0 we say finitary Remark (a) Te λ-ary morpisms are precisely te λ-presentable objects of te arrow category A. In contrast, M. Hébert introduced in [13] λ-presentable morpisms; tese are te morpisms f : A B wic are λ-presentable objects of te slice category A A. In te present paper we will not use te latter concept. (b) We work now wit te Finitary Injectivity Logic, i.e., te deduction system 2.4 applied to finitary morpisms. We generalize tis to te k-ary logic below Teorem Te Finitary Injectivity Logic is complete in every locally presentable category A. Tat is, given a set H of finitary morpisms in A, ten every finitary morpism wic is an injectivity consequence of H is a formal consequence in te deduction system 2.4. Briefly: H = implies H. Proof Given a finitary morpism : A B wic is an injectivity consequence of H, we prove tat H. (a) Te above object A as a weak reflection r : A A suc tat r is a transfinite composition of morpisms in Ĥ, see 3.3(a). Since H =, it follows tat A is injective w.r.t., wic yields a morpism u forming a commutative triangle r A A u B (b) Consider all commutative triangles as above were r : A A is any α-composite of morpisms in Ĥ for some ordinal α and u is arbitrary. We prove tat te least possible α is finite. Tis finises te proof of H : In case α = 0, we ave tat id = u, and we derive via identity and cancellation. In case α is a finite ordinal greater tan 0, we ave tat r is provable from H using pusout and composition. Consequently, via cancellation, we get. 13

14 Let C be te class of all ordinals α suc tat tere are an α-composite r of morpisms of Ĥ and a morpism u wit r = u. To sow tat te least member γ of C is finite, we prove tat for eac ordinal γ ω in C we can find anoter ordinal in C wic is smaller tan γ. A. Case γ = β + m, wit β a limit ordinal and m > 0 finite. Let a i,i+1 (i < β + m) be te corresponding cain wit r = a 0,β+m. Since a β,β+1 lies in Ĥ, we can express it as a pusout of some morpism k : D D in H: k D A 0 a 01 A 1... a 12 A i a i,i+1 D q p p A i+1 a i+1,i+2 A i+2... a i+2,i+3 A β aβ,β+1 A β+1 v i P i v i+1 v i+2 p i,i+1 P i+1 p i+1,i+2 P i+2 p i+2,i+3 v β... P β We ave a colimit A β = colim i<β A i of a cain of morpisms. Hence, because D is finitely presentable, p factorizes as p = a iβ q for some i < β and some morpism q : D A i. Let v i be a pusout of k along q, and form a sequence v j of pusouts of k along a ij q (j < β) as illustrated in te diagram above (taking colimits at te limit ordinals). Ten it is easily seen, due to p = a iβ q, tat v β = colim j<β v j is a pusout of k along p. Tus, witout loss of generality, P β = A β+1 and v β = a β,β+1. Observe tat, since a j,j+1 lies in Ĥ, pusout implies tat p j,j+1 Ĥ for all i j < β. Also v i Ĥ since it is a pusout of k along q. Consequently, a 0,β+1 is a β-composite of morpisms b j,j+1 (j < β) of Ĥ as follows (were l is te first limit ordinal after i): and b j,j+1 = a j,j+1 for all j < i, b i,i+1 = v i, b j,j+1 = p j 1,j for all i < j < l, b j,j+1 = p j,j+1 for all l j < β. Tus r = a 0,β+m is a (β + (m 1))-composite of morpisms of Ĥ. B. Case γ is a limit ordinal. Te morpism u : B A = colim i<γ A i factors, since B is finitely presentable, troug some a iγ, i < γ: u = a iγ u for some u : B A i. 14

15 Te parallel pair A = A 0 u A i a 0i is clearly merged by te colimit morpism a iγ of A γ = colim i<γ A i. Since A is finitely presentable, om(a, ) preserves tat colimit, consequently (see (ii) in 3.1.b), te parallel pair is also merged by a connecting morpism a ij : A i A j for some i < j < γ: Tis gives us a commutative triangle a ij u = a 0j. a 01 a A 0 12 A 1... Aj a ij u B tus a 0j is a j-composite of morpisms of Ĥ wit j < γ Remark Te above teorem immediatly generalizes to te k-ary Injectivity Logic, i.e., to te deduction system of 2.18 applied to k-ary morpisms. Recall tat for every set of objects in a locally presentable category tere exists a cardinal k suc tat all tese objects are k-presentable. Consequently, for every set H {} of morpisms tere exists k suc tat all members are k-ary. Te proof tat H = implies H is completely analogously to 3.6: We sow tat te least possible α is smaller tan k, tus in Cases A. and B. we work wit γ k Corollary Te Injectivity Logic is sound and complete in every locally presentable category. In fact, given H = find a cardinal k suc tat all members of H {} are k-ary morpisms. Ten is a formal consequence of H by Remark Te above corollary also follows from te Small Object Argument (see 3.3(a)): if : A B is an injectivity consequence of H and if r : A A is te corresponding weak reflection, ten r is clearly a formal consequence of H. Since A is injective w.r.t., it follows tat r factors troug, tus, is a formal consequence of r (via cancellation). 4 Strongly locally ranked categories 4.1. Remark Recall tat a factorization system in a category is a pair (E, M) of classes of morpisms containing all isomorpisms and closed under composition suc tat (a) every morpism f : A B as a factorization f = m e wit e : A C in E and m : C B in M 15

16 and (b) given anoter suc factorization f = m e tere exists a unique diagonal fill-in morpism d making te diagram commutative. A e C e d m C B m Te factorization system is called left-proper if every morpism of E is an epimorpism. In tat case te E-quotients of an object A are te quotient objects of A represented by morpisms of E wit domain A Definition Let (E, M) be a factorization system. We say tat an object A as M-rank λ, were λ is a regular cardinal, provided, tat (a) om(a, ) preserves λ-filtered colimits of diagrams of M-morpisms (i.e., given a λ-filtered diagram D wose connecting morpisms lie in M, ten every morpism f : A colimd factors, essentially uniquely, troug a colimit map of D) and (b) A as less tan λ E-quotients. If λ = ℵ 0 we say tat te object A as finite M-rank Examples (1) For te factorization system (Iso, All), rank λ is equivalent to λ- presentability. (2) In te category Top of topological spaces, coose (E, M) = (Epi, Strong Mono). Here te M-subobjects are precisely te embeddings of subspaces. Every topological space A of cardinality α as M-rank λ wenever λ > 2 2α. In fact, om(a, ) preserves λ-directed unions of subspaces since α < λ. And te amount of quotient objects of A (carried by epimorpisms) is at most β α E βt β were E β is te number of equivalence relations on A of order β and T β is te number of topologies on a set of cardinality β. Since E β and T β are bot 2 2β, we ave β α E βt β α 2 2α 2 2α < λ, tus we conclude tat A as less tan λ quotients Remark Every E-quotient of an object of M-rank λ also as M-rank λ. In fact (a) in 4.2 follows easily by diagonal fill-in, and (b) is obvious Definition A category A is called strongly locally ranked provided tat it as a left-proper factorization system (E, M) suc tat (i) A is cocomplete; (ii) every object as an M-rank, and all objects of te same M-rank form a set up to isomorpism; 16

17 (iii) for every cardinal µ te collection of all objects of M-rank µ is closed under E- quotients and under µ-small colimits, i.e., colimits of diagrams wit less tan µ morpisms; and and (iv) te subcategory of all objects of A and all morpisms of M is closed under filtered colimits in A. Remark Te statement (iv) means tat, given a filtered colimit wit connecting morpisms in M, ten (a) te colimit cocone is formed by morpisms of M (b) every oter cocone of M-morpisms as te unique factorizing morpism in M Examples (1) Every locally presentable category is strongly locally ranked: coose E isomorpisms, M all morpisms. In fact, see [4], 1.9 for te proof of (ii), wereas (iii) and (iv) old trivially. (2) Coose E epimorpisms, M strong monomorpisms. Here categories suc as Top (wic are not locally presentable) are included. In fact, for a space A of cardinality α we ave tat om(a, ) preserves λ-filtered colimits (=unions) of subspaces wenever α < λ. Tus, by coosing a cardinal λ > α bigger tan te number of quotients of A we get an M-rank of A. It is easy to verify (iii) and (iv) in Top. (3) Let B be a full, isomorpism closed, E-reflective subcategory of a strongly locally ranked category A. If B is closed under filtered colimits of M-morpisms in A, ten B is strongly locally ranked. In fact, B is closed under M in te sense tat given m : A B in M wit B B, ten A B. (Indeed, we ave a reflection r A : A A in E and m = m r A for a unique m ; tis implies tat r A E is an isomorpism, tus, A B.) Terefore te restriction of (E, M) to B yields a factorization system. It fulfils (ii)-(iv) of 4.5 because B is closed under filtered colimits of M-morpisms. (4) Te category Haus of Hausdorff spaces is strongly locally ranked: it is an epireflective subcategory of Top closed under filtered unions of subspaces Observation In a strongly locally ranked category te class M is closed under transfinite composition. Tis follows from (iv) Definition A morpism is called k-ary if its domain and codomain ave M-rank k. In case k = ℵ 0 we speak of finitary morpisms. 17

18 4.9. Remark Te name strongly locally ranked was cosen since our requirements are somewat stronger tan tose of [2]: tere a category is called locally ranked in case it is cocomplete, as an (E, M)-factorization, is E-cowellpowered and for every object A tere exists an infinite cardinal λ suc tat om(a, ) preserves colimits of λ-cains of M-monomorpisms. Our definition of rank and te condition 4.5(ii) imply tat te given category is E-cowellpowered. Tus, every strongly locally ranked category is locally ranked. An example of a locally ranked category tat is not strongly locally ranked is te category of σ-semilattices (posets wit countable joins and functions preserving tem): condition 4.5(iv) fails ere. Consider e.g. te ω-cain of te posets exp(n) (were n = {0, 1,..., n 1}), n ω, wit inclusion as order. Te colimit of tis cain is exp(n) ordered by inclusion. If M is te poset of all finite subsets of N wit an added top element, ten te embeddings exp(n) M form a cocone of te cain, but te factorization morpism exp(n) M is not a monomorpism. 5 A construction of weak reflections 5.1. Assumption In te present section A denotes a strongly locally ranked category. For every infinite cardinal k, A k denotes a cosen set of objects of M-rank k closed under E-quotients and k-small colimits. In particular, one may of course coose A k to be a set of representatives of all te objects of M-rank k up to isomorpism. Given a set H M of k-ary morpisms of A k (considered as a full subcategory of A), [2] provides a construction of a weak reflection in Inj H, wic generalizes te Small Object Argument (see 3.3). However, tis does not appear to be sufficient to prove our Completeness Teorem for te finitary case. Te aim of tis section is to present a different, more appropriate construction. We begin wit te case k = ω and come back to te general case at te end of tis section Convention (a) Morpisms wit domain and codomain in A ω are called petty. (b) Given a set H of petty morpisms, let H denote te closure of H under finite composition and pusout in A ω. (Tat is, H is te closure of H {id A ; A A ω } under binary composition and pusout along petty morpisms.) (c) Since H mora ω is a set, we can, for every object B of A ω, index all morpisms of H wit domain B by a set and tat indexing set can be cosen to be independent of B. Tat is, we assume tat a set T is given and tat for every object B A ω, is te set of all morpisms of H wit domain B. { B (t) : B B(t) ; t T } (5.1) 18

19 5.3. Diagram D A For every object A A ω we define a diagram D A in A and later prove tat a weak reflection of A in Inj H is obtained as a colimit of D A. Te domain D of D A, independent of A, is te poset of all finite words ε, M 1, M 1 M 2,..., M 1... M k (k < ω) were ε denotes te empty word and eac M i is a finite subset of T. Te ordering is as follows: M 1... M k N 1... N l iff k l and M 1 N 1,..., M k N k. Observe tat ε is te least element. We denote te objects D A (M 1... M k ) of te diagram D A by A M were M = M 1... M k, and if M 1... M k N 1... N l = N, we denote by a M,N : A M A N te corresponding connecting morpism of D A. We define tese objects and connecting morpisms by induction on te lengt k of te word M = M 1... M k considered. Case k = 0: A ε = A. Induction step: Assume tat all objects A M wit M of lengt less tan or equal to k and all connecting morpisms between tem are defined. For every word M of lengt k + 1 denote by M M te prefix of M of lengt k, and define te object A M as a colimit of te following finite diagram A K AK (t) A K (t) a K,M A M... were K ranges over all words K D wit K M and t ranges over te set M k+1. Tus, A M is equipped wit (te universal cone of) morpisms a M,M : A M A M (connecting morpism of D A ) and d K M(t) : A K (t) A M for all K M, t M k+1, 19

20 forming commutative squares a K,M A M A K AK (t) A K (t) (5.2) a M,M A M d K M (t) Tis defines te objects A M for all words of lengt k + 1. Next we define connecting morpisms a N,M : A N A M for all words N M. If te lengt of N is at most k, ten N M and we define a N,M troug te (already defined) connecting morpism a N,M by composing it wit te above a M,M. If N as lengt k + 1, we define a N,M as te unique morpism for wic te diagrams A K a K,N AK (t) A N A K (t) a N,N d K N (t) A N a N,M a d K N,M M (t) A M (K N, t N k+1 ) (5.3) commute. It is easy to verify tat te morpisms a N,M are well-defined and tat D A : D A preserves composition and identity morpisms Lemma All connecting morpisms of te diagram D A lie in H. Proof We first observe tat, given a finite diagram A i i B i f i C (i I) 20

21 wit all i in H, a colimit i A i B i f i d i C D (i I) (5.4) is obtained by first considering pusouts i of i along f i and ten forming a wide pusout of all i (i I). Consequently, te connecting morpisms of D A are formed by repeating one of te following steps: a finite wide pusout of morpisms in H, a composition of morpisms in H, and a pusout of a morpism in H along a petty morpism. Since H is closed, by 5.2, under te latter, it is closed under te first one in te obvious sense, see te construction of a finite wide pusout described in Example Lemma For every object A M of te diagram D A and every morpism : A M B of H tere exists a connecting morpism a M, N : A M A N of D A wic factors troug. Proof We ave M = M 1... M k and = AM (t) for some t T. Put N = M 1... M k {t}. Ten for K = M te definition of d K N (t) (see (5.2)) gives te following commutative diagram: Consequently, as required. A M id AM (t) A M (t) A M a M,N A N d K N (t) a M,N = d K N(t) AM (t) 5.6. Proposition Let H be a set of petty morpisms wit H M. Ten for every object A A ω a colimit γ M : A M  (M D) of te diagram D A yields a weak reflection of A in Inj H via r A = γ ε : A Â. Proof (1)  is injective w.r.t. H: We want to prove tat given H and f as follows B C f  ten f factors troug. Firstly, since  = colimd A is a directed colimit of H- morpisms (see 5.4) wit H M, and B as finite M-rank (because B A ω ), it follows 21

22 tat om(b, ) preserves te colimit of D A. Tus, tere exists a colimit morpism γ N : A N  troug wic f factors, f = γ N f. B C f f f  γ N A N A N (t) γ M A M a N, M By pusing H out along f we obtain a morpism H. Ten by 5.5 tere exists M N suc tat a N,M = for some : A N (t) A M. Te above commutative diagram proves tat f factors troug. (2) Let B be injective w.r.t. H. For every morpism f : A B we define a compatible cocone f M : A M B of te diagram D A by induction on k = te lengt of te word M suc tat f ε = f. Ten te desired factorization of f is obtained via te (unique) factorization g :  B wit g γ M = f M : in fact, g r A = f. For k k + 1, coose for every word N of lengt k and every t T a morpism f N (t) forming a commutative triangle A N AN (t) A N (t) f N f N (t) B (recalling tat B is H-injective because it is H-injective). Ten for every word M of lengt k+1 we ave a unique factorization f M : A M B making te following diagrams A K AK (t) A K (t) (5.5) a K, M d K M (t) a M,M f K (t) A M A M f M f M commutative for all K M and t M k+1. Let us verify te compatibility f M = f N a M, N for all M N in D. (5.6) Te last diagram yields f M = f M a M, M. Terefore, it is sufficient to prove (5.6) for words M and N of te same lengt k + 1. In order to do tat, we will sow tat f M d K M(t) = f N a M,N d K M(t), for all K M and t M k+1, (5.7) 22 B

23 and Concerning (5.7), we ave f M d K M (t) f M a M,M = f N a M,N a M,M. (5.8) = f K(t) = f N d K N (t), by replacing M by N in (5.5) = f N a M,N d K M (t), by (5.3). As for (5.8), we ave f M a M,M = f M = f N a M,N = f N a N,N a M,N, by replacing M by N in (5.5) = f N a M,N a M,M Convention Generalizing te above construction from ω to any infinite cardinal k, we call te morpisms of A k k-petty. Let us now denote by H k te closure of H under k-composition (2.10) and pusout in A k. Following 2.18, H k is closed under k-wide pusout. We again assume tat a set T is given suc tat, for every object B A k we ave an indexing B (t) : B B(t), t T of all morpisms of H k wit domain B Diagram D A Te poset D of 5.3 is generalized to a poset D k : Let P k T be te poset of all subsets of T of cardinality < k. Te elements of D k are all functions M : λ P k T were λ < k is an ordinal, including te case ε : 0 P k T. Te ordering is as follows: for N : λ P k T put M N iff λ λ and M i N i for all i < λ. We define, for every A A k, te diagram D A : D k A. Te objects D A (M) = A M and te connecting morpisms a M,N : A M A N (M N) are defined by transfinite induction on λ < k. For λ = 0 we ave A ε = A. Te isolated step is precisely as in 5.3, were for M : λ + 1 P k T we denote by M : λ P k T te domain-restriction. Te limit steps are defined via colimits of smoot cains, see 2.10: if λ < k is a limit ordinal and M : λ P k T is given, ten A M is a colimit of te cain A M/i (i < λ), were M/i is te domain restriction of M to i, wit te connecting morpisms a M/i, M/j : A M/i 23

24 A M/j for all i j < λ. Te proof tat tese cains are smoot is an easy transfinite induction. It is also easy to see tat all te above results old: Â = colimd A is an H-injective weak reflection of A, and all connecting morpisms of D A are members of H. Consequently, te proof of te following proposition is analogous to tat of 5.6: 5.9. Proposition Let H be a set of k-petty morpisms wit H k M. Ten for every object A A k a colimit γ M : A M Â of D A yields a weak reflection of A in InjH via r A = γ ε : A Â. 6 Completeness in strongly locally ranked categories 6.1. Assumption Trougout tis section A denotes a strongly locally ranked category. We first prove te completeness of te finitary logic. Recall tat te finitary morpisms are tose were te domain and codomain are of finite M-rank. Let us remark tat wenever te class M is closed under pusout, ten te metod of proof of Teorem 3.6 applies again. However, tis excludes examples suc as Haus (were strong monomorpisms are not closed under pusout) Teorem Te Finitary Injectivity Logic is complete in every strongly locally ranked category. Tat is, given a set H of finitary morpisms, every finitary morpism wic is an injectivity consequence of H is a formal consequence (in te deduction system of 2.4). Sortly: H = implies H Remark We do not need te full strengt of weak local presentation for tis result. We are going to prove te completeness under te following milder assumptions on A: (i) A is cocomplete and as a left-proper factorization system (E, M); (ii) A ω is a set of objects of finite M-rank, closed under finite colimits and E-quotients; (iii) M is closed under filtered colimits in A (see 4.5 (iv)). Te statement we prove is, ten, concerned wit petty morpisms (see 5.2). We sow tat for every set H of petty morpisms we ave H = implies H (for all petty). Te coice of A ω as a set of representatives of all objects of finite M-rank yields te statement of te teorem. Proof of 6.2 and 6.3 Let ten H be a set of petty morpisms, and let H denote te closure of H as in 5.2. (1) We first prove tat te teorem olds wenever H M. Moreover, we will sow tat for every petty injectivity consequence H = we ave a formal proof of from 24

25 assumptions in H suc tat te use of pusout is always restricted to pusing out along petty morpisms. To prove tis, consider, for te given petty injectivity consequence : A B of H, te weak reflection r A : A  in Inj H of 5.6. Te object  is injective w.r.t., tus r A factors troug via some f : B Â: r A A B f A M γ M a N, M  γ N Since B A ω, it as finite M-rank, and 5.4 implies tat om(b, ) preserves te colimit  = colimd A. Ten f factors troug one of te colimit morpisms γ N : A N Â: g A N f = γ N g for some g : B A N. We know tat r A = γ ε is te composite of te connecting morpism a ε, N : A A N of D A and γ N, terefore, γ N a ε, N = r A = γ N g. Tat is, te colimit morpism γ N merges te parallel pair a ε, N, g : A A N. Now te domain A as finite M-rank, tus om(a, ) also preserves  = colimd A. Consequently, by (ii) in 3.1(b) te parallel pair is also merged by some connecting morpism a N, M : A N A M of D A : a N, M a ε, N = a N, M g : A A M. Te left-and side is simply a ε, M, and tis is a morpism of H, see Lemma 5.4. Recall tat te definition of H implies tat every morpism in H can be proved from H using Finitary Injectivity Logic in wic pusout is only applied to pusing out along petty morpisms. Tus, we ave a proof of te rigt-and side a N, M g. Te last step is deriving from tis by cancellation. (2) Assuming H E, ten we prove tat Inj H is a reflective subcategory of A, and for every object A A ω te reflection map r A : A  is a formal consequence of H lying in E: H r A and r A E. In fact, from H E it follows tat H E (since E is closed under composition and pusout). Since A as only finitely many E-quotients, see 4.2, we can form a finite wide pusout, r A : A Â, of all E-quotients of A lying in H. Clearly, H r A, in fact, r A H. Te object  is injective w.r.t. H: given : P P in H and f : P Â, form a pusout of along f. Tis is an E-quotient in H, ten te same is true for 25

26 r A. Consequently, r A factors troug r A, and te factorization, i : B Â, is an epimorpism split by, tus, f = i g : P P f A r A Â Te morpism r A is a weak reflection: given a morpism u from A to an object C of Inj H, ten u factors troug r A because C is injective w.r.t. H and r A H. (3) Let H be arbitrary. We begin our proof by defining an increasing sequence of sets E i E of petty morpisms (i Ord). For every member f : A B of H we denote by f i a reflection of f in Inj E i : r A i A f B Â fi ˆB First step: E 0 = {id A ; A A ω }. Here Inj E 0 = A, tus f 0 = f. Isolated step: For eac f H, let f i = f i f i be te (E, M)-factorization of te reflection f i of f in Inj E i, and put and r B B E i+1 = E i {f i; f H}. Limit step: E j = i<j E i for limit ordinals j. We prove tat for every ordinal i we ave g H f i for every f H (6.1) Inj H = Inj E i Inj{f i } f H. (6.2) For i = 0, (6.1) and (6.2) are trivial (use cancellation for (6.1) and identity for (6.2)). Given i > 0, assuming tat H f j for all j < i, wit f : A B in H, tat is, H E i, we ave, by (2), tat H r B (6.3) were r B is te reflection of B in Inj E i. Tus, H f i r A. Moreover, r A is an epimorpism, terefore te following square A r A Â f i ˆB r A Â f i ˆB id is a pusout, wic proves H f i (via pusout). H f i ten follows by cancellation. 26

27 To prove (6.2), observe tat (6.1) implies Inj H Inj E i, and our previous argument yields Inj H Inj {f i } f H. Tus, it remains to prove te reverse inclusion: every object X injective w.r.t. E i {f i } f H is injective w.r.t. H. In fact, given f : A B in H and a morpism u : A X, ten since X Inj E i we ave a factorization u = v r A, and ten te injectivity of X w.r.t. f i yields te desired factorization of u troug f. A u f B r A X v  ˆB f i r B (4) Since A ω is a small category, tere exists an ordinal j wit We want to apply (1) to te category and te set E j = E j+1. A = Inj E j, A ω = A ω obja. Let us verify tat A satisfies te assumptions (i) (iii) of Remark 6.3 w.r.t. E = E mora and M = M mora. Ad(i): A is cocomplete because it is reflective in A. Moreover, since te reflection maps lie in E, it follows tat (E, M ) is a factorization system: in fact, A is closed under factorization in A. Since E Epi(A), we ave E Epi(A ). Ad(iii): It is sufficient to prove tat A is closed under filtered colimits of M - morpisms in A. In fact, let D be a filtered diagram in A wit connecting morpisms in M, and let c t : C t C (t T ) be a colimit of D in A. Ten C A, i.e., C is injective w.r.t. f j :  E for every f H. Tis follows from  aving finite M-rank (because A A ω implies  A ω due to te fact tat r A : A  is an E-quotient): since om(â, ) preserves te colimit of D, every morpism u :  C factors troug some of te colimit morpisms:  v C t c t C u f j E Since C t A is injective w.r.t. f j, we ave a factorization of v troug f j, and terefore, u also factors troug f j. Tis proves C A. 27

28 Ad(ii): Due to te above, every object of A aving a finite M-rank in A as a finite M -rank in A. Also, a finite colimit of objects of A in A is a reflection (tus, an E-quotient) of te corresponding finite colimit in A. Tus, it lies in A ω. Next we claim tat te set H = {f j ; f H} fulfils H M and H is closed under petty identities, composition, and pusouts along petty morpisms. In fact, in te above (E, M)-factorization of f j : A f B r A  f j f j f j D ˆB r B we know tat f j lies in E j+1 = E j and  is injective w.r.t. E j, tus, f j is a split monomorpism (as well as an epimorpism, since E Epi(A)). Tus, f j is an isomorpism, wic implies f j M. H contains id A for every A A ω because H contains it; H is closed under composition because H is (and f f j is te action of te reflector functor from A to Inj E j ). Finally, H is closed under pusout along petty morpisms. In fact, to form a pusout of f j :  ˆB along u :  C in A = Inj E j, we form a pusout, g, of f along u r A in A, and compose it wit te reflection map r D of te codomain D: A f B r A r B  f j ˆB v ˆv u ˆD ĝ r D C g D Since C lies in A, we can assume r C = id C, and te reflection ĝ = r D g of g in A is ten a pusout of f j along u. Now f H implies g H, and we ave ĝ = g j H. (5) We are ready to prove tat if a petty morpism : A B is an injectivity consequence of H, ten H in A. We write H A for te latter since we work witin two categories: wen we apply (1) to A we use A for formal consequence in A. Analogously wit = A and = A. Let ĥ :  ˆB be a reflection of in A, ten H = A ĥ because every object C A = Inj E j wic is injective w.r.t. H = {f j } f H is, due to (6.2), injective w.r.t. H in A. Ten C is injective w.r.t., and from C A it follows 28

29 easily tat C is injective w.r.t. ĥ. Due to (4) we can apply (1). Terefore, H A ĥ. We tus ave a proof of ĥ from H in A. We modify it to obtain a proof of from H in A. We ave no problems wit a line of te given proof tat uses one of te assumptions f j H : we know from (6.1) tat H A f j, and we substitute tat line wit a formal proof of f j in A. No problem is, of course, caused by te lines using composition or cancellation. But we need to modify te lines using pusout because A is not closed under pusout in A. However, a pusout, g, of a morpism g along a petty morpism u in A u P g Q P g r Q g Q is obtained from a pusout, g, of g along u in A by composing it wit a reflection map r Q of te pusout codomain. Recall tat P, P, Q A ω imply Q A ω. Tus, we can replace te line g of te given proof by using pusout in A (deriving g ), followed by a proof of r Q (recall from (6.3) tat H A r Q ) and an application of composition. We tus proved tat H A ĥ. Since r B = ĥ r A and H A r A (see (6.3)), we conclude H A ĥ r A ; by cancellation ten H A Corollary (Compactness Teorem) Let H be a set of finitary morpisms in a strongly locally ranked category. Every finitary morpism wic is an injectivity consequence of H is an injectivity consequence of a finite subset of H Remark We proceed by generalizing te completeness result from finitary to k- ary, were k is an arbitrary infinite cardinal. Te k-ary logic, ten, deals wit k-ary morpisms (i.e., tose aving bot domain and codomain of M-rank k) and te k-ary Injectivity Deduction System of Teorem Te k-ary Injectivity Logic is complete in every strongly locally ranked category. Tat is, given a set H of k-ary morpisms, ten every k-ary morpism wic is an injectivity consequence of H is a formal consequence (in te k-ary Injectivity Deduction System). Proof Te wole proof is completely analogous to tat of Teorem 6.2. As described in Remark 6.3 we work under te following milder assumptions on te category A: (i) A is cocomplete and as a left-proper factorization system (E, M); (ii) A k is a set of objects of M-rank k, closed under colimits of less tan k morpisms and under E-quotients; 29