aa r X i v : . [ m a t h . A T ] J u l REPRESENTABILITY THEOREMS, UP TO HOMOTOPY
DAVID BLANC AND BORIS CHORNY
Abstract.
We prove two representability theorems, up to homotopy, forpresheaves taking values in a closed symmetric combinatorial model category V . The first theorem resembles the Freyd representability theorem, the secondtheorem is closer to the Brown representability theorem. As an application wediscuss a recognition principle for mapping spaces. Introduction
It is a classical question in homotopy theory whether for a given space X ∈ T op ∗ there exists a space Y ∈ T op ∗ such that X ≃ hom( S , Y ). Several solutions to thisquestion have emerged, beginning with Sugawara’s work in [26]. The approachproposed by Stasheff in [25], Boardman and Vogt in [5], and May in [20] are knownnowadays as operadic, while Segal’s loop space machine (see [24]), is closer toLawvere’s notion of an algebraic theory. Later on, the canonical delooping machineby Badzioch, Chung, and Voronov (see [2]) provided a simplicial algebraic theory T n , n ≥ n -fold loop space as a homotopy algebraover T n .It is natural to ask whether there exist similar recognition principles for mappingspaces of the form X = hom( A, Y ) for A ∈ T op ∗ other than S n ? When A = S ∨ S ,for example, there is no simplicial algebraic theory T such that all spaces of theform X = hom( S ∨ S , Y ) = Ω Y × Ω Y are homotopy algebras over T (see [2,p. 2]). More generally, if a space A has rational homology in more than one positivedimension, there is no simplicial algebraic theory T such that all the spaces of theform X = hom( A, Y ) have the structure of homotopy algebras over T , [3].In this paper we suggest using a larger category than an algebraic theory forthe recognition of arbitrary mapping spaces, up to homotopy. This category will beclosed under arbitrary homotopy colimits, unlike an algebraic theory which is closedonly under finite (co)products. The minimal subcategory of spaces containing A and closed under the homotopy colimits is denoted C ( A ). This approach was firstintroduced by Badzioch, Dorabia la, and the first author in [1], where an attemptto limit the homotopy colimits involved was made. We take a different approachhere, and consider the functors defined on the large subcategory of spaces C ( A ).Given a space X ∈ T op ∗ , suppose there exists a functor F : C ( A ) → T op ∗ takinghomotopy colimits to homotopy limits and satisfying F ( A ) ≃ X . The question of Date : July 25, 2019.1991
Mathematics Subject Classification.
Primary 55U35; Secondary 55P91, 18G55.
Key words and phrases.
Mapping spaces,representable functors,Bousfield localization, non-cofibrantly generated, model category.The first author acknowledges the support of ISF 770/16 grant.The second author acknowledges the support of ISF 1138/16 grant. whether there exists a space Y such that X ≃ hom( A, Y ) for some Y is equivalentto the question of representability of the functor F , up to homotopy.Theorems about representability of functors are naturally divided into two maintypes: Freyd and Brown representability theorems. In both cases some exactnesscondition for the functor under consideration is necessary. Theorems of Freyd typeuse a set-theoretical assumption about the functor: for example, the solution setcondition, or accessibility (see [14, 3, Ex. G,J], [17, 4.84], and [22, 1.3]). Theoremsof Brown type use set-theoretical assumptions about the domain category, such asthe existence of sufficiently many compact objects (see [6], [21], [18], and [13]).In this paper we address the question of representability (up to homotopy) offunctors taking values in a closed symmetric combinatorial model category. Aftersome technical preliminaries in Section 2, we prove the Freyd version of repre-sentability up to homotopy in Section 3. The solution set condition is replaced bythe requirement that the functor be small. Theorem 3.1 generalizes [17, Theorem4.84] to functors defined on a V -model category, rather than just a V -category. Atthe same time, it generalizes a result the first author on the representability of smallcontravariant functors from spaces to spaces (see [8]).The Brown version of representability up to homotopy is proved in Section 4. Theset-theoretical condition concerning the domain category is local presentability. Inother words, we show that for any V -presheaf H defined on a combinatorial V -modelcategory M and taking homotopy colimits to homotopy limits, there is a fibrantobject Y ∈ M and a natural transformation h : H ( − ) → hom( − , Y ), which is aweak equivalence for every cofibrant X ∈ M . A similar theorem for functors takingvalues in simplicial sets was proved by Jardine in [16]. However, the conditionsrequired for the Brown representability, up to homotopy, to hold are formulatedfor the homotopy category of the model category and do not allow for an easyverification in an arbitrary combinatorial model category.In Section 5 we provide an example of a non-small presheaf, defined on a non-combinatorial model category, which is not representable up to homotopy, Thisshows that representability theorems are not tautological.In Section refrecognition we interpret Brown representability up to homotopy asa recognition principle for mapping spaces for an arbitrary space A (rather thanjust S n ). 2. (Model) Categorical preliminaries For every closed symmetric monoidal combinatorial model category V and V -model category M (not necessarily combinatorial), we can consider the categoryof small presheaves V M op . This is a V -category of functors, which are left Kanextensions from small subcategory of M . The category V M op is cocomplete by[17, 5.34]. Since V is a combinatorial model category, it is in particular locallypresentable. Therefore, the category of small presheaves V M op is also complete by[10]. For a V -category C we denote by C the underlying category of C (enrichedonly in Set). Definition 2.1.
A natural transformation f : F → G in (cid:0) V M op (cid:1) is called a cofibrant-projective weak equivalence (respectively, a cofibrant-projective fibration ) EPRESENTABILITY THEOREMS, UP TO HOMOTOPY 3 if for all cofibrant M ∈ M , the induced map f M : F ( M ) → G ( M ) is a weak equiv-alence (respectively, a fibration). The notion of a projective weak equivalence (re-spectively, a projective fibration ) in (cid:0) V M op (cid:1) is a particular case of the cofibrant-projective analog, when all objects of M are cofibrant – e.g., for the trivial modelstructure on M . If (cofibrant-)projective fibrations and weak equivalences give riseto a model structure on the category of small presheaves, this model structure iscalled (cofibrant-)projective .First of all, we would like to establish the existence of the cofibrant-projectivemodel structure on (cid:0) V M op (cid:1) . For a simplicial model category M , with V = S (thecategory of simplicial sets), this was proven in [9, 2.8]. For a combinatorial modelcategory M op , this was proven in [4, 3.6]. But the case of contravariant smallfunctors from a combinatorial model category to V is not covered by the previousresults. Condition 2.2.
Every trivial fibration in V is an effective epimorphism in V (cf., [23, II, p. 4.1] ). Basic examples of categories satisfying this condition are (pointed) simplicialsets, spectra, and chain complexes.
Theorem 2.3.
Let V be a closed symmetric monoidal model category satisfyingcondition 2.2, and M a V -model category. The category of small functors (cid:0) V M op (cid:1) may be equipped with the cofibrant-projective model structure. Moreover, V M op be-comes a V -model category.Proof. Let I and J be the classes of generating cofibrations and generating trivialcofibrations in V , and let M cof denote the subcategory of cofibrant objects of M .The cofibrant-projective model structure on the category of small presheaves V M op is generated by the following classes of maps. I = { R M ⊗ i | I ∋ i : A ֒ → B, M ∈ M cof } and J = { R M ⊗ j | J ∋ j : U ˜ ֒ → V, M ∈ M cof } where R M is the representable functor X hom( X, M ) ∈ V .It suffices to verify that these two classes of maps admit the generalized smallobject argument,[7]. More specifically, we need to show that I and J are locallysmall. In other words, for any map f : X → Y we need to find a set W of maps in I -cof (respectively, J -cof), such that every morphism of maps R M ⊗ i → f factorsthrough an element in W . By adjunction, it is sufficient to find a set of cofibrantobjects U , such that every map R M → X A × Y A Y B factors though an element of U . Consider the functor F = X A × Y A Y B . Like any small functor, F : M op → V is a left Kan extension from a small full subcategory D of M , hence a weightedcolimit of representable functors, which, in turn, may be viewed as a coequalizer,by the dual of [17, 3.68]: F = Z D R D ⊗ F D = coeq a f : D ′ → D R D ′ ⊗ F D ⇒ a D R D ⊗ F D . DAVID BLANC AND BORIS CHORNY
Therefore, every V -natural transformation R A → F in (cid:0) V M op (cid:1) factors through R D ⊗ F D for some D ∈ D by the weak Yoneda lemma for V -categories. Unfortu-nately, R D ⊗ F D is not necessarily I -cofibrant. However, we can find an I -cofibrantobject U having a factorization R A → U → R D ⊗ F D for every cofibrant A ∈ M .Let q : ˜ D ˜ ։ D be a cofibrant replacement in M , and U := R ˜ D ⊗ ˜ D , with the map U → R D ⊗ F D composed of R D ⊗ q and hom( − , q ) ⊗ ˜ D . It suffices to show thatthe induced map hom( R A , U ) → hom( R A , R D ⊗ F D ) is an epimorphism. This mapfactors, in turn, as a composition of two maps hom( A, ˜ D ) ⊗ ˜ D → hom( A, D ) ⊗ ˜ D → hom( A, D ) ⊗ D , each of which is given by tensoring an effective epimorphism withan object of V . A map is an effective epimorphism if and only if it is the coequalizerof some pair of parallel maps (see, e.g., the dual of [15, 10.9.4]). Hence, this is acomposition of two effective epimorphisms, and so an epimorphism. In particular,if S is a unit of V , then hom V ( S, hom( R A , U )) → hom V ( S, hom( R A , R D ⊗ F D ))is a surjection of sets. V M op becomes a V -model category by [4, Prop. 3.18]. (cid:3) Definition 2.4.
Consider the following classes of maps in V M op : F = (cid:26) ∅ = hocolim ∅ ∅ → R hocolim ∅ ∅ = R ∅ (cid:27) F = { R X ⊗ A → R X ⊗ A | X ∈ M , A ∈ V – cofibrant objects } . where the map R X ⊗ A → R X ⊗ A is the unit of the adjunction Y : M ⇆ V M op : − ⊗ Id M . F = hocolim R A / / (cid:15) (cid:15) R B R C → R D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A / / (cid:15) (cid:15) B (cid:15) (cid:15) C / / D – homotopy pushout ofcofibrant objects in M F = hocolim k<κ ( R A → . . . → R A k → R A k +1 → . . . ) (cid:15) (cid:15) R colim k<κ A k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ֒ → . . . ֒ → A k ֒ → A k +1 ֒ → . . . ,A k cofibrant for all k < κ Set F := F ∪ F ∪ F ∪ F . Definition 2.5.
Given a class F of natural transformations of cofibrant-projectivelycofibrant small functors M op → V (for M and V as in Theorem 2.3), we say that afunctor F : C → V is F - local if every f : G → H in F induces a weak equivalence f ∗ : hom( H, F ) → hom( G, F ). Proposition 2.6.
Let M be a V -model category, and assume that the underlyingcategory of the category of small functors V M op may be equipped with the cofibrantprojective model structure. Let F ∈ V M op be a cofibrant-projectively fibrant smallfunctor taking weighted homotopy colimits of cofibrant objects to homotopy limitsin V . Then the functor F is F -local (with respect to the class F of maps fromDefinition 2.4).Proof. This follows from Yoneda’s lemma and the fact that the colimit of a sequenceof cofibrations of cofibrant objects is a homotopy colimit. (cid:3)
EPRESENTABILITY THEOREMS, UP TO HOMOTOPY 5
Remark . We have only included in the class F those morphisms which arerequired for the proof of the inverse implication: F -local functors are equivalentto the representable functors (see Theorem 3.1). In some situations the class F ofmaps may be reduced even further. For example, if V = S , then the subclass F ofmaps is redundant, since every weighted homotopy colimit in a simplicial categorycan be expressed in terms of the classical homotopy colimits (cf. [8, Lemma 3.1]).If V = M = Sp for some closed symmetric monoidal combinatorial model of spec-tra, again F is not needed, by to Spanier-Whitehead duality (see [4, Lemma 7.2]).However, in general weighted homotopy colimits cannot be expressed in terms ofclassical homotopy colimits (that is, homotopy colimits with contractible weight),as is shown by Luk´aˇs-Vokˇr´ınek in [27].3. Freyd representability theorem, up to homotopy
Theorem 3.1.
Let V be a closed symmetric monoidal combinatorial model categorysatisfying condition 2.2, and suppose that the domains of the generating cofibrationsof V are cofibrant. If M is a V -model category, then the underlying category of thecategory V M op of small V -presheaves on M may be equipped with the cofibrant-projective model structure. A small functor F is cofibrant-projectively weakly equiv-alent to a representable functor if and only if it takes homotopy colimits of cofibrantobjects to homotopy limits.Proof. Let λ be a regular cardinal such that V is a λ -combinatorial model cate-gory. Let I and J be its sets of generating cofibrations and trivial cofibrations,respectively. We assume that the domains of the maps in I are cofibrant.Given a small functor F ∈ V M op taking homotopy colimits to homotopy limits, let e F ˜ ։ F be a cofibrant replacement for F in the cofibrant-projective model structure.Then there is a λ -sequence F → F → · · · → F k → F k +1 → · · · ˜ F such that F ( M ) = ∅ for all M ∈ M , e F = colim F k , and F k +1 is obtained from F k as apushout R M ⊗ A / / (cid:127) _ (cid:15) (cid:15) F k (cid:15) (cid:15) R M ⊗ B / / F k +1 , where M ∈ M is cofibrant, and ( A ֒ → B ) ∈ I , A , B ∈ V are also cofibrant byassumption.Our proof will proceed by induction. Recall the class F of maps from Defini-tion 2.4, and note that the fibrant replacement of the given functor F is F -local byProposition 2.6.Note also that F = ∅ is F -equivalent to R ∅ = R hocolim ∅ ∅ , since the map ∅ → R ∅ is in F ⊂ F .Suppose by induction that F k is F -equivalent to a representable functor R X k ,where X k is a fibrant and cofibrant object of M . There is then a commutative DAVID BLANC AND BORIS CHORNY diagram R X ′ k "b " " " " ❊❊❊❊❊❊❊❊ R M ⊗ A (cid:15) (cid:15) / / ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ R X k R M ⊗ A / / (cid:127) _ (cid:15) (cid:15) ε e e ❑❑❑❑❑❑❑❑❑❑ F k (cid:15) (cid:15) ; ; ①①①①①①①① O O R M ⊗ B / / y y ssssssssss F k +1 R M ⊗ B where the upper horizontal arrow is induced by the universal property of the unit ofthe adjunction, and X ′ k is a fibrant and cofibrant object of M obtained as a middleterm of the factorization M ⊗ A ֒ → X ′ k ˜ ։ X k .Since the functor F k is cofibrant, there exists a lift F k → R X ′ k which is an F -equivalence by the 2-out-of-3 property and does not violate the commutativity ofthe above diagram, since the upper slanted arrow R M ⊗ A → R X ′ k is also a naturalmap induced by the universal property of the unit of adjunction ε .Let P := M ⊗ B ` M ⊗ A X ′ k . We then obtain a commutative diagram R M ⊗ A (cid:15) (cid:15) / / R ′ X k (cid:15) (cid:15) R M ⊗ A / / (cid:127) _ (cid:15) (cid:15) e e ❑❑❑❑❑❑❑❑❑❑ F k (cid:15) (cid:15) < < ①①①①①①①① R M ⊗ B / / y y ssssssssss F k +1 ❋❋❋❋ R M ⊗ B / / R P in which all the solid slanted arrows are F -equivalences and the inner square isa homotopy pushout. Hence, the homotopy pushout of the outer square is F -equivalent to F k +1 , and thus the dashed arrow is also an F -equivalence. Finally,let X k +1 denote the middle element in the factorization X ′ k ֒ → X k +1 ˜ ։ ˆ P . Then R P → R ˆ P is also an F -equivalence, and thus by the 2-out-of-3 property, so is thelift F k +1 → R X k +1 (which exists since F k +1 cofibrant).If κ is a limit ordinal, then F κ = colim k<κ F k . Since this is the colimit of asequence of cofibrations of cofibrant functors, colim k<κ F k is the homotopy colimithocolim k<κ F k . However, F k is F -equivalent to R X k . Moreover, by constructionthere is a sequence of cofibrations X ֒ → . . . ֒ → X k ֒ → X k +1 ֒ → . . . . Hencehocolim R X k is F -equivalent to R colim X k .If κ is not large enough to ensure that colim k<κ X k is fibrant, we can considerthe fibrant replacement colim k<κ X k ˜ ֒ → \ colim k<κ X k = X κ . Combining these facts EPRESENTABILITY THEOREMS, UP TO HOMOTOPY 7 together we conclude that F κ is F -equivalent to a representable functor R X κ , rep-resented by a fibrant and cofibrant object.For κ large enough we have F = F κ , so F is F -equivalent to a functor representedby a fibrant and cofibrant object. But F is an F -local functor by Proposition 2.6,and so is R X κ for every κ . Hence, F ≃ R X κ for some κ , since an F -equivalence of F -local functors is a weak equivalence. (cid:3) The formal category theoretic dual of Theorem 3.1 is the following:
Theorem 3.2.
Let V be a closed symmetric monoidal combinatorial model categorysatisfying condition 2.2, and let M be a V -model category such that the category (cid:0) V M (cid:1) may be equipped with the fibrant-projective model structure. A small functor F : M op → V is then fibrant-projectively weakly equivalent to a representable functorif and only if it takes homotopy limits of fibrant objects to homotopy limits. Brown representability theorem, up to homotopy
In this section we prove a homotopy version of the Brown Representability The-orem for contravariant functors from a locally presentable V -model category M to V taking homotopy colimits to homotopy limits.Note that our proof does not use explicitly the presence of compact objects, asmost known proofs in this field do. Rather we show directly that a contravarianthomotopy functor from M to V is cofibrant projectively weakly equivalent to a smallfunctor, and then use the Freyd representability theorem, up to homotopy. Lemma 4.1.
Let M be a combinatorial V -category. Then any functor H : M op → V taking homotopy colimits of cofibrant objects to homotopy limits is cofibrant-projectively weakly equivalent to a small homotopy functor F ∈ V M op .Proof. Suppose M is a λ -combinatorial model category, for some cardinal λ suchthat the weak equivalences are a λ -accessible subcategory of the category of mapsof M .Consider a functor H : M op → V taking homotopy colimits of cofibrant objectsto homotopy limits, with the natural map H → F = Ran i : M λ ֒ → M i ∗ H , where Ran is the right Kan extension along the inclusion of categories i : M λ ֒ → M .Here M λ denotes the full subcategory of λ -presentable objects in M .This map is a weak equivalence in the cofibrant-projective model category, be-cause both functors take homotopy colimits to homotopy limits and every cofibrantobject of M is a ( λ -filtered) homotopy colimit of λ -presentable cofibrant objects,[19, Cor. 5.1], on which the two functors coincide.But we can interpret the right Kan extension as a weighted inverse limit ofrepresentable functors F = Ran i : M λ ֒ → M i ∗ H = Z M ∈ M λ hom( H ( M ) , M ) = { i ∗ H, i ∗ Y } , where Y : M → V M op is the Yoneda embedding and { i ∗ H, i ∗ Y } is the weightedinverse limit of i ∗ Y indexed by i ∗ H (see [17, 3.1]).Then by the theorem of Day and Lack in [10], F is small as an inverse limit ofsmall functors. (cid:3) DAVID BLANC AND BORIS CHORNY
Remark . When we speak about sufficiently large filtered colimits in a combi-natorial model category, they turn out to be homotopy colimits, no matter if theobjects participating in them are cofibrant or not. Therefore the above Lemmaadmits the following concise formulation: any functor taking homotopy colimits tohomotopy limits is levelwise weakly equivalent to a small functor.
Theorem 4.3.
Let V be a closed symmetric monoidal model category satisfyingcondition 2.2, and M a combinatorial V -model category. Any functor H : M op → V taking homotopy colimits of cofibrant objects to homotopy limits is then cofibrant-projectively weakly equivalent to a representable functor.Proof. Follows from Theorem 3.1 by Lemma 4.1. (cid:3) Counter-example
We have proved so far two representability theorems, up to homotopy. The firstis of Freyd type, i.e., some set theoretical conditions are required to be satisfied bythe functor in question. The second one is of Brown type, i.e., some set theoreticalassumptions apply to the domain category. But what happens if we make noset theoretical assumptions on either the domain category or the functor? Areexactness conditions enough to ensure representability up to homotopy?Mac Lane’s classical (folklore) example of a functor B : Grp → Set, which as-signs to each group G the set of all homomorphisms from the free product of alarge collection of non-isomorphic simple groups to G , is an example of a (strictly)non representable functor. Notice that neither does B satisfy the solution set con-dition, nor is the category Grp op locally presentable. Perhaps representability upto homotopy is less demanding and would persist without any conditions?Our example is similar in nature to Mac Lane’s example, but it has also an-other predecessor: in [11], Dror-Farjoun gave an example of a failure of Brownrepresentability for generalized Bredon cohomology.Consider the closed symmetric combinatorial model category of spaces S , with M = S S the category of small functors. Let B : S → S be a functor which isnot small, such as B = hom(hom( − , S ) , S ). Note that B / ∈ M , since B is notaccessible and all small functors are. However, H = hom( − , B ) : M → S is welldefined, since for any small functor F ∈ M , there exists a small subcategory i : A ֒ →S such that F = Lan i i ∗ F and hence H ( F ) = hom( F, B ) = hom S A ( i ∗ F, i ∗ B ) ∈ S .We have defined a functor H : M op → S taking homotopy colimits of cofibrantobjects to homotopy limits, but it is not representable, even up to homotopy:otherwise, there would exist a small fibrant functor A ∈ M such that H ( − ) ≃ hom( − , A ). By J. H. C. Whitehead’s argument we know A ≃ B , but then B wouldpreserve the λ -filtered homotopy colimits for some λ . This is a contradiction, since B does not preserve filtered colimits even of discrete spaces.6. Mapping space recognition principle
In this section we assume that the closed symmetric monoidal combinatorialmodel category V is the category T op ∗ of ∆-generated topological spaces. This isa locally presentable version of the category of topological spaces, first proposed byJeff Smith and described in detail by Fajstrup and Rosicky in [12].Let X ∈ T op ∗ be a path-connected space, and let A ∈ T op ∗ be a CW-complex.We will describe a sufficient condition for there to exist a space Y ∈ T op ∗ such EPRESENTABILITY THEOREMS, UP TO HOMOTOPY 9 that X ≃ hom( A, Y ). For example, if A = S , the required condition is that X canbe equipped with an algebra structure over the little intervals operad. In practicethat means that the space X admits k -ary operations, i.e., maps X × . . . × X → X satisfying a long list of higher associativity conditions.For a space A more general then S it is insufficient to consider the structuregiven by the maps X × . . . × X → X , by [3]. We will consider, instead, the structuregiven by the mutual interrelations of all possible homotopy inverse limits of X –the structure we would have if X indeed was equivalent to hom( A, Y ) for some Y .Indeed, consider the subcategory of A -cellular spaces C ( A ) ⊂ T op ∗ . This is theminimal subcategory containing A and closed under the homotopy colimits. Anyhomotopy colimit of a diagram involving A is then taken by the functor hom( − , Y )into the homotopy limit of an opposite diagram involving X . In other words, everymapping space X is equipped with a functor F X : C ( A ) → T op ∗ . Moreover, thisfunctor has a very nice property: it takes homotopy colimits into homotopy limits.Our goal is to show the converse statement: if for a given X ∈ T op ∗ there existsa functor F : C ( A ) op → T op ∗ taking homotopy colimits to homotopy limits andsatisfying F ( A ) ≃ X , this F is weakly equivalent to a representable functor, i.e.,there is a space Y such that F ( − ) ≃ hom( − , Y ). Theorem 6.1.
For any cofibrant A ∈ T op ∗ and any X ∈ T op ∗ , there is anobject Y ∈ T op ∗ satisfying X ≃ hom( A, Y ) if and only if there exists a functor F : C ( A ) op → T op ∗ taking homotopy colimits to homotopy limits and satisfying F ( A ) ≃ X .Proof. Necessity of the condition is clear. We will prove the sufficiency now.Consider the right Bousfield localization of T op ∗ with respect to A . By [15,5.1.1(3)] we obtain a cofibrantly generated model structure, which is also combina-torial, since we have chosen to work with a locally presentable model of topologicalspaces. We denote the new model category by T op A ∗ . The subcategory of cofi-brant objects of T op A ∗ is then C ( A ) as above. We denote the inclusion functor by i : C ( A ) → T op A ∗ .Given a functor F : C ( A ) → T op ∗ taking homotopy colimits to homotopy limitsand satisfying F ( A ) = X , consider the left Kan extension H = Lan i F of F alongthe inclusion i . This functor H : T op A ∗ → T op ∗ then satisfies the condition ofTheorem 4.3, hence there exists Y ∈ T op A ∗ such that H ( − ) ≃ hom( − , Y ), inparticular, H ( A ) = F ( A ) ≃ X ≃ hom( A, Y ) . (cid:3) References [1] B. Badzioch, D. Blanc, and W. Dorabia l a. Recognizing mapping spaces.
J. Pure Appl.Algebra , 218(1):181–196, 2014.[2] B. Badzioch, K. Chung, and A. A. Voronov. The canonical delooping machine.
J. Pure Appl.Algebra , 208(2):531–540, 2007.[3] B. Badzioch and W. Dorabia l a. A note on localizations of mapping spaces.
Israel J. Math. ,177:441–444, 2010.[4] G. Biedermann and B. Chorny. Duality and small functors.
Algebr. Geom. Topol. , 15(5):2609–2657, 2015.[5] J. M. Boardman and R. M. Vogt.
Homotopy invariant algebraic structures on topologicalspaces . Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, Berlin-New York, 1973.[6] E. H. Brown, Jr. Cohomology theories.
Ann. of Math. (2) , 75:467–484, 1962.[7] B. Chorny. A generalization of Quillen’s small object argument.
Journal of Pure and AppliedAlgebra , 204:568–583, 2006. [8] B. Chorny. Brown representability for space-valued functors.
Israel J. Math. , 194(2):767–791,2013.[9] B. Chorny. Homotopy theory of relative simplicial presheaves.
Israel J. Math. , 205(1):471–484, 2015.[10] B. Day and S. Lack. Small limits of functors.
Journal of Pure and Applied Algebra , 210:651–663, 2007.[11] E. Dror Farjoun. Homotopy and homology of diagrams of spaces. In
Algebraic topology (Seat-tle, Wash., 1985) , Lecture Notes in Math. 1286, pages 93–134. Springer, Berlin, 1987.[12] L. Fajstrup and J. Rosicky. A convenient category for directed homotopy.
Theory and Appli-cations of Categories , 21(1):7–20, 2008.[13] J. Franke. Brown representability theorem for triangulated categories.
Topology , 40(4):667–680, 2001.[14] P. Freyd.
Abelian categories . Harper and Row, New York, 1964.[15] P. S. Hirschhorn.
Model categories and their localizations , volume 99 of
Mathematical Surveysand Monographs . American Mathematical Society, Providence, RI, 2003.[16] J. F. Jardine. Representability theorems for presheaves of spectra.
J. Pure Appl. Algebra ,215(1):77–88, 2011.[17] G. M. Kelly.
Basic concepts of enriched category theory , volume 64 of
London MathematicalSociety Lecture Note Series . Cambridge University Press, Cambridge, 1982.[18] H. Krause. A brown representability theorem via coherent functors.
Topology , 41:853–861,2002.[19] M. Makkai, J. Rosick´y, and L. Vokˇr´ınek. On a fat small object argument.
Adv. Math. , 254:49–68, 2014.[20] J. P. May.
The geometry of iterated loop spaces . Springer-Verlag, Berlin-New York, 1972.Lectures Notes in Mathematics, Vol. 271.[21] A. Neeman.
Triangulated categories , volume 148 of
Annals of Mathematics Studies . PrincetonUniversity Press, 2001.[22] A. Neeman. Brown representability follows from rosicky’s theorem.
Journal of Topology ,2:262–276, 2009.[23] D. G. Quillen.
Homotopical Algebra . Lecture Notes in Math. 43. Springer-Verlag, Berlin,1967.[24] G. Segal. Categories and cohomology theories.
Topology , 13:293–312, 1974.[25] J. D. Stasheff. Homotopy associativity of H -spaces. I, II. Trans. Amer. Math. Soc. 108 (1963),275-292; ibid. , 108:293–312, 1963.[26] M. Sugawara. h -spaces and spaces of loops. Mathematical Journal of Okayama University ,5:5–11, 1956.[27] L. Vokˇr´ınek. Homotopy weighted colimits. Preprint, 2012.
Department of Mathematics, University of Haifa, Haifa, Israel
E-mail address : [email protected] Department of Mathematics, University of Haifa at Oranim, Tivon, Israel
E-mail address ::