Theory and Applications of Categories, Vol. 20, No. 2, 2008, pp. 18–24.

GENERALIZED BROWN REPRESENTABILITY IN HOMOTOPY CATEGORIES: ERRATUM

JIŘÍ ROSICKÝ

Abstract. Propositions 4.2 and 4.3 of the author’s article (Theory Appl. Categ. 14 (2005), 451-479) are not correct. We show that their use can be avoided and all remaining results remain correct.

Propositions 4.2 and 4.3 of the author’s [3] are not correct and I am grateful to J. F. Jardine for pointing it out. In fact, consider the diagram D sending the one morphism category to the point Δ_{0} in the homotopy category Ho(SSet) of simplicial sets. The standard weak colimit of D is the standard weak coequalizer

\xymatrix{Δ_{0} \ar[r]^{id} \ar[r]_{id} & Δ_{0}}

This weak coequalizer is the homotopy pushout

\xymatrix{Δ_{0} \ar[r] & P \\ Δ_{0} ⨿ Δ_{0} \ar[u] \ar[r] & Δ_{0} \ar[u]}

But P is weakly equivalent to the circle and thus it is not contractible. Hence the standard weak colimit of D is not weakly equivalent to the colimit of D.

A fatal error is in the last line of part I of the proof of Proposition 4.2. We could neglect the identity morphisms in the construction of weak colimits via coproducts and weak coequalizers. It means that we take the coproduct

∐_{e:d→d′} Dd

indexed by all non-identity morphisms e of D. Then, taking for D the three-element chain and for D : D → SSet the constant diagram at Δ_{0}, the modified standard weak colimit of D is not contractible again. In fact, R. Jardine showed that for each small category D and each diagram D : D → SSet constant at Δ_{0}, the weak standard colimit of D is weakly equivalent to the graph of D. The graph of the three-element chain 0 < 1 < 2 contains the boundary of the simplex Δ_{1} but not Δ_{1} itself. It does not seem possible to find a modification of standard weak colimits satisfying Proposition 4.2.

Received by the editors 2008-01-10.

Transmitted by Ross Street. Published on 2008-01-20.

2000 Mathematics Subject Classification: 18G55, 55P99.

Key words and phrases: Quillen model category, Brown representability, triangulated category, accessible category.

\begin{matrix}c \\ ◯ \end{matrix} Jiří Rosický, 2008. Permission to copy for private use granted.

These incorrect results were used only for proving Theorems 5.4 and 5.7 (numbered references here and below are for results in [3]). Fortunately, we can avoid their use. Theorem 5.4 says that the functor

E_{λ} : Ho(K) → Ind_{λ}(Ho(K_{λ}))

is essentially surjective on objects, i.e., that for each X in Ind_{λ}(Ho(K_{λ})) there is K in Ho(K) with E_{λ}K ≅ X. Theorem 5.7 then adds that E_{λ} is full. This formulation is not correct and one has to replace it by E_{λ} being essentially surjective in the sense that every morphism f : X → Y in Ind_{λ}(Ho(K_{λ})) is isomorphic to E_{λ}(g) in the category of morphisms of Ind_{λ}(Ho(K_{λ})) for some g : K → L. The latter means the commutativity of a square

\xymatrix{E_{λ}K \ar[r]^{E_{λ}(g)} & E_{λ}L \\ X \ar[u]^{≅} \ar[r]_{f} & Y \ar[u]_{≅}}

An essentially surjective functor is essentially surjective on objects (by using f = id_{X}). Thus the correct formulation of Theorems 5.4 and 5.7 is the following statement.

1.1. Theorem. Let K be a locally λ-presentable model category satisfying the conditions (G_{λ}^{1}) and (G_{λ}^{2}). Then the functor

E_{λ} : Ho(K) → Ind_{λ}(Ho(K_{λ}))

is essentially surjective.

Proof. Since (G_{λ}^{1}) implies (G_{λ}^{4}) (by Remark 3.4), it follows from Corollary 5.2 that it suffices to prove that Ind_{λ} P_{λ} is essentially surjective. At first, we prove that it is essentially surjective on objects. We start in the same way as in the proof of Theorem 5.4. We consider an object X in Ind_{λ}(Ho(K_{λ})) and express it as a λ-filtered colimit (δ_{d} : Dd → X) of the canonical diagram D : D → Ho(K_{λ}). Without any loss of generality, we can assume that Dd belong to K_{cf} for all d in D. Then we take a standard weak colimit (\overline{δ}_{d} : \overline{D}d → K) of the lifting \overline{D} of D along E_{λ}. Standard weak colimits are given by a construction in K and let \overline{K} be the resulting object. Thus K = P \overline{K} and we can assume that \overline{K} is in K_{cf}. In K, λ-filtered colimits commute both with coproducts and with homotopy pushouts. The second claim follows from the fact that homotopy pushouts are constructed via pushouts and (cofibration, trivial fibration) factorizations and the latter preserve λ-filtered colimits (by (G_{λ}^{1})). Consequently, λ-filtered colimits commute in K with the construction of standard weak colimits and thus \overline{K} is a λ-filtered colimit α_{E} : \overline{K}_{E} → \overline{K} of objects \overline{K}_{E} giving standard weak colimits of λ-small subdiagrams D_{E} : E → Ho(K_{λ}) of D. All objects \overline{K}_{E} belong to K_{λ}. We can even assume that each E has a terminal object d_{E}. Let u_{E} : Dd_{E} → P_{λ}\overline{K}_{E} be the corresponding component of a standard weak colimit cocone. Clearly, these morphisms form a natural transformation from D to the the diagram consisting of P_{λ}\overline{K}_{E}. Since d_{E} is a terminal object of E, there is a morphism s_{E} : P_{λ}\overline{K}_{E} → Dd_{E} with s_{E}u_{E} = id_{Dd_{E}}. Thus each morphism u_{E} is a split monomorphism. Thus the colimit

u : X → E_{λ}\overline{K} = (Ind_{λ} P_{λ})\overline{K}

of E_{λ}(u_{E}) is a λ-pure monomorphism (see [1]). This argument is a correct part of the proof of Proposition 4.3.

Now, consider a pullback

\xymatrix{K_{λ} \ar[r]^{P_{λ}} & Ho(K_{λ}) \\ \overline{\overline{D}} \ar[u]^{\overline{\overline{D}}} \ar[r]_{\overline{P}} & D \ar[u]_{D}}

We will show that the functor \overline{P} : \overline{\overline{D}} → D is final. Observe that, for each object \overline{d} in \overline{\overline{D}}, we have \overline{P} (\overline{d}) = QR\overline{d} and the same for morphisms. For each object d in D, there is \overline{d} in \overline{\overline{D}} with d = \overline{P} (\overline{d}). Consider two morphisms f_{1} : d → \overline{P} (\overline{d}_{1}) and g_{1} : d → \overline{P} (\overline{d}_{2}) in D. Since D is λ-filtered, there is a commutative square (where we replace \overline{P} by QR)

\xymatrix{QR\overline{d}_{1} \ar[r]^{h_{2}} & e \\ d \ar[u]^{f_{1}} \ar[r]_{g_{1}} & QR\overline{d}_{2} \ar[u]_{h_{1}}}

in D. Let α : Id_{K} → R_{f} and β : R_{c} → Id_{K} be natural transformation given by fibrant and cofibrant replacements. Since α is a pointwise trivial cofibration and β is a pointwise trivial fibration, both QRα and QRβ are natural isomorphisms. Thus we get the following zig-zags in the comma category d ↓ \overline{P}

\xymatrix{f_{1} & f_{2} \ar[l]_{QRβd̄_{1}} \ar[r]^{QRα_{Rcd̄_{1}}} & f_{3} \ar[r]^{QRh_{2}} & f_{4}}

and

\xymatrix{g_{1} & g_{2} \ar[l]_{QRβd̄_{2}} \ar[r]^{QRα_{Rcd̄_{2}}} & g_{3} \ar[r]^{QRh_{1}} & g_{4}}

Here, f_{2} = QR(β_{\overline{d}_{1}})^{−1} · f_{1} and f_{3}, f_{4} are the corresponding compositions; analogously for g_{1}. Since f_{4} = g_{4}, f_{1} and g_{1} are connected by a zig-zag in the category d ↓ \overline{P}. Thus the functor \overline{P} is final. Consequently,

X ≅ colim D ≅ colim(\overline{P} · \overline{\overline{D}}).

The latter object is isomorphic to

(Ind_{λ} P_{λ}) colim \overline{\overline{D}}

provided that the diagram \overline{\overline{D}} is λ-filtered. Evidently, this diagram is λ-filtered when X ≅ (Ind_{λ} P_{λ})L for some L in K. Thus X belongs to the essential image of Ind_{λ} P_{λ} if and only if the corresponding diagram \overline{\overline{D}}_{X} is λ-filtered (the index X denotes that \overline{\overline{D}} belongs to X).

Consider X and a λ-small subcategory A of \overline{\overline{D}}_{X}. Since the functor P_{λ} preserves λ-small coproducts, we can assume that A consists of morphisms

h_{i} : \overline{\overline{D}}_{X}d_{1} → \overline{\overline{D}}_{X}d_{2},

i ∈ I where card I < λ. Then h_{i}, i ∈ I are morphisms uδ_{d_{1}} → uδ_{d_{2}} in the diagram \overline{\overline{D}}_{Y} for Y = (Ind_{λ} P_{λ})\overline{K}. Since the latter diagram is λ-filtered, there are f : P_{λ}A → Y and h : \overline{\overline{D}}_{X}d_{2} → A such that h coequalizes all h_{i}, i ∈ I and fP_{λ}(h) = uδ_{d_{2}}. Since u is λ-pure, there is g : P_{λ}A → X such that gP_{λ}(h) = δ_{d_{2}}. Thus h : δ_{d_{2}} → g coequalizes h_{i}, i ∈ I in \overline{\overline{D}}_{X}. We have proved that \overline{\overline{D}}_{X} is λ-filtered and thus

X ≅ (Ind_{λ} P_{λ}) colim \overline{\overline{D}}_{X}.

We have thus proved that Ind_{λ} P_{λ} is essentially surjective on objects.

Now, consider a morphism h : X → Y in Ind_{λ}(Ho(K_{λ})) and express X and Y as canonical λ-filtered colimits (δ_{Xd} : D_{X}d → X) and (δ_{Y d} : D_{Y} d → Y) of objects from Ho(K_{λ}). Let \overline{\overline{D}}_{X} : \overline{\overline{D}}_{X} → K_{λ} and \overline{\overline{D}}_{Y} : \overline{\overline{D}}_{Y} → K_{λ} be the diagrams constructed above. The prescription Hf = hf yields a functor

H : D_{X} → D_{Y}

such that D_{Y} H = D_{X}. Since the diagrams \overline{\overline{D}}_{X} and \overline{\overline{D}}_{Y} are given by pullbacks, there is a functor

\overline{H} : \overline{\overline{D}}_{X} → \overline{\overline{D}}_{Y}

such that \overline{\overline{D}}_{Y} \overline{H} = \overline{\overline{D}}_{X}. This gives a morphism

\overline{h} : colim \overline{\overline{D}}_{X} → colim \overline{\overline{D}}_{Y}

such that (Ind_{λ} P_{λ})\overline{h} ≅ h. Thus the functor Ind_{λ} P_{λ} is essentially surjective.

We say that a locally λ-presentable model category K is essentially λ-Brown provided that the functor E_{λ} is essentially surjective. Thus the correct formulation of Corollary 5.8 is as follows.

1.2. Corollary. Let K be a locally λ-presentable model category satisfying the conditions (G_{λ}^{1}) and (G_{λ}^{2}). Then K is essentially λ-Brown.

Thus, for a combinatorial model category K, there are arbitrarily large regular cardinals λ such that K is essentially λ-Brown. Recall that K is λ-Brown if E_{λ} is essentially surjective on objects and full. The original formulation of Corollary 5.9 remains true for stable model categories.

1.3. Proposition. Let K be a locally λ-presentable model category satisfying the conditions (G_{λ}^{1}), (G_{λ}^{2}) and such that E_{λ} reflects isomorphisms. Then K is λ-Brown.

Proof. Consider a morphism h : E_{λ}P K_{1} → E_{λ}P K_{2}. In the proof of the Theorem above, we have found a morphism \overline{h} : L_{1} → L_{2} such that E_{λ}P \overline{h} ≅ h. Following the construction of objects L_{1} and L_{2}, there are morphisms t_{i} : K_{i} → L_{i}, i = 1, 2, such that

(E_{λ}P (t_{1}), E_{λ}P (t_{2})) : h → E_{λ}P \overline{h}

is an isomorphism. The reason is that the canonical diagram of K_{i} with respect to K_{λ} is a subdiagram of the diagram \overline{\overline{D}}_{i} constructed in the proof above and L_{i} = colim \overline{\overline{D}}_{i}, i = 1, 2. Since E_{λ} reflects isomorphisms, P (t_{1}) and P (t_{2}) are isomorphisms. We get the morphism

h′ = P (t_{2})^{−1}P (\overline{h}t_{1}) : P K_{1} → P K_{2}

with E_{λ}(h′) = h. Thus E_{λ} is full.

1.4. Corollary. Let K be a combinatorial stable model category. Then there are arbitrarily large regular cardinals λ such that K is λ-Brown.

Proof. It follows from Proposition 6.1, the proof of Proposition 6.4 and the Proposition above.

1.5. Remark. 5.11 (3) implies a correct substitute of Proposition 4.3 saying that, for a stable locally λ-presentable model category K satisfying the conditions (G_{λ}^{1}) and (G_{λ}^{2}), the functor P : K → Ho(K) sends λ-filtered colimits of λ-presentable objects to minimal weak λ-filtered colimits.

The distinction between being Brown and essentially Brown is important, which is manifested by the following strengthening of Proposition 6.11.

1.6. Proposition. Let K be a model category which is λ-Brown for arbitrarily large regular cardinals λ. Then idempotents split in Ho(K).

Proof. Let f : P K → P K be an idempotent in Ho(K). There is a regular cardinal λ such that K is λ-Brown and K is λ-presentable in K. Since idempotents split in Ind_{λ}(Ho(K_{λ})), there are morphisms p : E_{λ}K → E_{λ}L and u : E_{λ}L → E_{λ}K such that pu = id_{E_{λ}} and E_{λ}f = up. Since E_{λ} is full, there are morphisms \overline{p} : K → L and \overline{u} : L → K with E_{λ}\overline{p} = p and E_{λ}\overline{u} = u. Since E_{λ} is faithful on Ho(K_{λ}), we have pu = id_{K} and f = up. Hence idempotents split in Ho(K).

Now, Remark 6.12 implies that SSet cannot be λ-Brown for arbitrarily large regular cardinals λ.

The proof of Proposition 6.10 is lacking a verification that each object A from Ho(K_{μ}) is μ-small. Also, the proof that Ho(K_{μ}) is μ-perfect has to be corrected because the coproduct of fibrant objects does not need to be fibrant and its fibrant replacement destroys the coproduct structure in K. I am grateful to B. Chorny for bringing these gaps to my attention. In what follows, a corrected proof of Proposition 6.12 is presented. Its first two paragraphs and the last one remain unchanged. This means that we are replacing the third paragraph of the proof.

We will show that each object A from Ho(K_{μ}) is μ-small. Recall that, following Remark 2.4 (2), K satisfies the conditions (G_{μ}^{i}) for i = 1, 2, as well, and thus it is μ-Brown (see the Corollary above). Consider a morphism

f : A → ∐_{i∈I} K_{i}

in Ho(K). We have

∐_{i∈I} K_{i} ≅ colim ∐_{j∈J} K_{j}

where the colimit is taken over all subsets J ⊆ I having cardinality smaller than μ. Since

E_{μ} colim ∐_{j∈J} K_{j} ≅ colim E_{μ} ∐_{j∈J} K_{j}

(see 6.5), E_{μ}f factorizes through some E_{μ} ∐_{j∈J} K_{j}. Thus f factorizes through some ∐_{j∈J} K_{j}.

It remains to show that Ho(K_{μ}) is μ-perfect. Consider a morphism f : A → ∐_{i∈I} K_{i} in

Ho(K) where card I < μ. Each K_{i}, i ∈ I, is a minimal μ-filtered colimit

(k_{j}^{i} : A_{ij} → K_{i})_{j∈J_{i}}

of objects A_{ij} ∈ Ho(K_{μ}). Hence all subcoproducts X = ∐_{i∈I} A_{ij_{i}} belong to Ho(K_{μ}). Let

Z be their minimal weak μ-filtered colimit. Since minimal weak colimits commute with coproducts, Z ≅ ∐_{i∈I} K_{i}.

Thus E_{μ}f factorizes through some E_{μ}X of some subcoproduct X and therefore f factorizes through X, which yields [2], 3.3.1.2 in the definition of perfectness.

References

[1] J. Adámek and J. Rosický, Locally Presentable and Accessible Categories, Cambridge University Press 1994.

[2] A. Neeman, Triangulated Categories, Princeton Univ. Press 2001.

[3] J. Rosický, Generalized Brown representability in homotopy categories, Theory and Applications of Categories 14(2005), 451–479.

Department of Mathematics Masaryk University 602 00 Brno, Czech Republic

Email: rosicky@math.muni.cz

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