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The Orthogonal Subcategory Problem and the Small Object Argument

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The Orthogonal Subcategory Problem and the Small Object Argument The Orthogonal Subcategory Problem and the Small Object Argument Ji·r¶³Ad¶amek,y Michel H¶ebert and Lurdes Sousaz May 8, 2007 Abstract 1 Introduction Our paper is devoted to two classical problems of category theory: the Orthogonal Subcat- egory Problem which asks, given a class H of morphisms, whether the subcategory H? of all objects orthogonal to every member of H is reflective. And the Small Object Argument which asks whether the subcategory Inj H of all objects injective to every member of H is weakly reflective and, moreover, the weak reflection maps can be chosen to be cellular (that is: they lie in the closure of H under trans¯nite composition and pushout). The orthogonal subcategory problem was solved for cocomplete categories A with a factorization system (E; M) satisfying some mild side conditions by Peter Freyd and Max Kelly [7]: the answer is a±rmative whenever the class H is "almost small" by which we mean that H is a union of a set and a subclass of E. We call categories satisfying the conditions formulated by Freyd and Kelly locally bounded. Example: the category Top of topological spaces is locally bounded for (Epi; StrongMono). This is, of course, the best factorization system for the above result. Also the category of topological spaces with an additional binary relation is locally bounded; here we present a class for which the orthogonal subcategory is not reflective. Thus, the orthogonal subcategory problem does not work for arbitrary classes. This contrasts to the result of Ji·r¶³Rosick¶yand the ¯rst author that in locally presentable categories the orthogonal subcategory problem has, assuming the large cardinal Vop·enka's Principle, always an a±rmative answer, see [5]. Now locally presentable categories are locally bounded for (StrongEpi; Mono), which is a weaker factorization system for our purposes; it does not seem to be known whether they are bounded for (Epi; StrongMono) in general. We prove that, nevertheless, independent of set theory, in locally presentable categories yFinancial support by the Ministry of Education of the Czech Republic, Project MSM 6840770014 is acknowledged. zFinancial support by Centre for Mathematics of University of Coimbra and by School of Technology of Viseu is acknowledged. 1 the result of Freyd and Kelly extends to classes almost all members of which are (just) epimorphisms; and also to classes consisting of ¸-presentable morphisms (in the sense of [10]) and strong epimorphisms. Our most surprising counter-example concerns the category UCPO of unary algebras de¯ned on cpo's: we present a category in which UCPO is the orthogonal subcategory H? where all members of H but one are epimorphisms, but UCPO fails to be (even weakly) reflective - however, the larger category is locally ranked in the sense of [4]. This demonstrates that, although the de¯nitions of locally bounded and locally ranked are almost identical, their properties can di®er dramatically. In our previous work [2] we have formulated a logic for orthogonality for deriving the \orthogonality consequence" of a class H of morphisms. This means those morphisms k such that every object orthogonal to all members of H is also orthogonal to k. We prove that this logic is complete in all the above cases yielding the a±rmative answer of the orthogonal subcategory problem. The Small Object Argument is an important classical result of homotopy theory formu- lated by D. Quillen [15], see also P. Gabriel and M Zisman [8]. In its generalized form it states that for every set H of morphisms the category Inj H allows for H-cellular weak reflec- tions. This is true in every locally ranked category, as proved in [4]. Our example UCPO mentioned above demonstrates that here we cannot generalize to almost small classes H. However, in locally bounded categories we prove that the Small Object Argument holds for all almost small classes. Under the Vop·enka's Principle, this holds in locally presentable categories for classes almost all members of which are (just) epimorphisms. It is an open problem whether this result can be achieved without a large-cardinal axiom. Grigori Ro»suformulated a logic for injectivity [16] in the case of ¯nitary epimorphisms which we generalized in [1] to a logic of injectivity for arbitrary morphisms. The completeness result for that logic we proved is improved in the present paper: besides covering the above cases for which the existence of H-cellular weak reflections is established, we also prove the completeness in locally presentable categories for classes made of ¸-presentable morphisms and strong epimorphisms. This contrasts to the non completeness for classes of morphisms in general, proved in [2], independently of set theory. 2 Locally Bounded and Locally Ranked Categories Throughout we work in categories in which each object is \¸-small" for some in¯nite regular cardinal ¸. In the case of locally presentable categories \¸-small" objects are just the ¸- presentable ones. But we consider other types of smallness de¯ned in this section. 2.1. De¯nitions Let K be a cocomplete category with a proper factorization system (E; M), ¸ an in¯nite regular cardinal, and K an object of K. (1) K is said to have bound ¸, if hom(K; ¡) preserves ¸-directed unions of M-subobjects. Explicitly: given a ¸-directed collection of subobjects` mi : Mi ! M (i 2 I) in M whose union is M in the sense that the morphism [mi]: i2I Mi ! M lies in E, then every morphism from K to M factors through mi for some i 2 I. (2) K is said to have rank ¸, if hom(K; ¡) preserves ¸-directed colimits of diagrams whose connected morphisms lie in M. 2 2.2. De¯nition A category K together with a proper factorization system (E; M) is called locally bounded provided that (i) K is cocomplete and E-cowellpowered and (ii) every object has a bound. 2.3. De¯nition If in the above de¯nition we replace (ii) by (ii)0 every object has a rank. we obtain the de¯nition of locally ranked category. 2.4. Remark Locally bounded categories were considered by Freyd and Kelly in [7] for the study of the Orthogonal Subcategory Problem, although they did not give them a name. Locally ranked categories were introduced in [4] and showed to be suitable as a setting for a generalized Small-Object Argument. There are locally ranked categories which are not locally bounded (see Example 3.7), but we do not know if the converse implication is veri¯ed. 2.5. Examples (a) All locally presentable categories are both locally bounded and locally ranked w.r.t. the factorization system (StrongEpi; Mono) ([7], [5]). (b) The category Top of topological spaces and continuous functions is locally bounded w.r.t. the factorization system (Surjections; Embeddings) (cf. [7]). In fact, strong monomor- phisms are just inclusions of subspaces. It follows that every topological space of cardinality less than ¸ (a regular cardinal) is bounded by ¸. Moreover Top is locally ranked w.r.t. the same factorization system (see [4]). (c) The category Haus of Hausdor® topological spaces is also locally bounded w.r.t. the factorization system (Surjections; Embeddings). In fact, the same is true for every epireflective subcategory of Top. This follows from the local boundedness of Top and the fact that every one of its epireflective subcategories is closed under embeddings. 2.6. De¯nition In locally bounded categories w.r.t. a factorization system (E; M), a class of morphisms will be called almost small if its members which do not lie in E form just a set. 3 The Orthogonal Subcategory Problem The Orthogonal Subcategory Problem (OSP for short) asks wether given a collection H of morphisms then the orthogonality class H?, which is the full subcategory of all objects orthogonal to every member of H, is reflective. From Max Kelly (10.1 and 10.2 of [11]), we know that the OSP has an a±rmative answer: (1) for sets of morphisms in locally ranked categories, and (2) for almost small classes of morphisms in locally bounded categories. In what concerns locally presentable categories, we know from [5] that under the Vop·enka's Principle the reflectivity of H? holds for general classes of morphisms. In [2] we an- swered a±rmatively (without the assumption of any large-cardinal set-theoretic axiom) to the OSP for classes of ¸-presentable morphisms and regular epimorphisms. On the other 3 hand, since, as observed in [7], locally presentable categories are locally bounded w.r.t. the (StrongEpi; Mono) factorization system, (2) above gives an a±rmative answer for classes consisting of strong epimorphisms plus a set of any morphisms. Here we go further: OSP has an a±rmative answer for classes whose non epimorphic members form a set, and also for classes made of ¸-presentable morphisms and strong epimorphisms. The main result of this section enlightens the picture of locally ranked categories and lo- cally bounded categories. In fact, as far as we know, until now the question of the reflectivity of H? in locally ranked categories for almost small classes of morphisms remained open. We will show that, in spite of the similarity of locally bounded and locally ranked categories, there appears to be a basic di®erence: we present a locally ranked category with an almost small class H of morphisms such that H? is not reflective. 3.1. Theorem In every locally presentable category the subcategory H? is reflective for any class H = Hs [He of morphisms where Hs is small and He consists of epimorphisms. ? Proof From [5] we know that Hs is a reflective subcategory in the given category A.
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