Math. J. Okayama Univ. 57 (2015), 79–84

ON MODEL STRUCTURE FOR COREFLECTIVESUBCATEGORIES OF A MODEL CATEGORY
Tadayuki Haraguchi

1. Introduction
Let C be a coreflective subcategory of a cofibrantly generated model category D. In this paper we show that under suitable conditions C admits acofibrantly generated model structure which is left Quillen adjunct to themodel structure on D. As an application, we prove that well-known convenient categories of topological spaces, such as k-spaces, compactly generatedspaces, and ∆-generated spaces [3] (called numerically generated in [12]) admit a finitely generated model structure which is Quillen equivalent to thestandard model structure on the category Top of topological spaces.
2. Coreflective subcategories of a model category
Let D be a cofibrantly generated model category [7, 2.1.17] with generating cofibrations I, generating trivial cofibrations J and the class of weakequivalences WD . If the domains and codomains of I and J are finite relativeto I-cell [7, 2.1.4], then D is said to be finitely generated.
Recall that a subcategory C of D is said to be coreflective if the inclusion functor i : C → D has a right adjoint G : D → C, so that there is anatural isomorphism ϕ : HomD (X, Y ) → HomC (X, GY ). The counit of thisadjunction ǫ : GY → Y (Y ∈ D) is called the coreflection arrow.
Theorem 2.1. Let C be a coreflective subcategory of a cofibrantly generated
model category D which is complete and cocomplete. Suppose that the unit of
the adjunction η : X → GX is a natural isomorphism, and that the classes Iand J of cofibrations and trivial cofibrations in D are contained in C. ThenC has a cofibrantly generated model structure with I as the set of generating
cofibrations, J as the set of generating trivial cofibrations, and WC as theclass of weak equivalences, where WC is the class of all weak equivalences
contained in C. If D is finitely generated, then so is C. Moreover, theadjunction (i, G, ϕ) : C → D is a Quillen adjunction in the sense of [7,1.3.1].
Proof. It suffices to show that C satisfies the six conditions of [7, 2.1.19]
with respect to I, J and WC . Clearly, the first condition holds because
Mathematics Subject Classification. Primary 55U40; Secondary 55U35.
Key words and phrases. model category, Quillen equivalence, numerically generated
space.
79

80

TADAYUKI HARAGUCHIWC satisfies the two out of three property and is closed under retracts. To
see that the second and the third conditions hold, let IC -cell and JC -cellbe the collections of relative I-cell and J-cell complexes contained in C,
respectively. Since IC -cell and JC -cell are subcollections of the collectionsof relative I-cell and J-cell complexes in D, respectively, the domains of I
and J are small relative to IC -cell and JC -cell, respectively. The rest of theconditions are verified as follows. Let f : X → Y be a map in C. Sinceη : X → GX is isomorphic for X ∈ D, f is I-injective in C if and only
if it is I-injective in D. Similarly, f is J-injective in C if and only if it
is J-injective in D. Let f be an I-cofibration in D. Then it has the left
lifting property with respect to all I-injective maps in C. Hence f is anI-cofibration in C. Conversely, let f be an I-cofibration in C. Suppose weare given a commutative diagram
X −−−−→ ApfyyY −−−−→ B
where p is I-injective in D. Then there is a relative I-cell complex g : X → Z[7, 2.1.9] such that f is a retract of g by [7, 2.1.15]. Since g is an I-cofibrationin D, there is a lift Z → A of g with respect to p. Then the composite
Y → Z → A is a lift of f with respect to p. Therefore f is an I-cofibrationin D. Similarly, f is a J-cofibration in C if and only if it is a J-cofibrationin D. Thus we have the desired inclusions• JC -cell ⊆ WC ∩ IC -cof,• IC -inj ⊆ WC ∩ JC -inj, and• either WC ∩ IC -cof ⊆ JC -cof or WC ∩ JC -inj ⊆ IC -inj.
Here IC -inj and IC -cof denote, respectively, the classes of I-injective maps
and I-cofibrations in C, and similarly for JC -inj and JC -cof. Therefore C
is a cofibrantly generated model category by [7, 2.1.19].
It is clear, by the definition, that C is finitely generated if so is C.
Finally, to prove that (i, G, ϕ) is a Quillen adjunction, it suffices to showthat G : D → C is a right Quillen functor, or equivalently, G preserves
J-injective maps in D by [7, 1.3.4] and [7, 2.1.17]. Let p : X → Y be a
J-injective map in D. Suppose there is a commutative diagram
A −−−−→ GX



Gpyfy
B −−−−→ GY

COREFLECTIVE SUBCATEGORIES OF A MODEL CATEGORY

81

where f ∈ J. Then we have a commutative diagramǫA −−−−→ GX −−−−→ X
pfyyǫ

B −−−−→ GY −−−−→ Y.
Since p is J-injective in D, there is a lift h : B → X of f . Thus we have a liftGh ◦ η : B ∼= GB → GX of f with respect to Gp. Therefore Gp : GX → GY
is J-injective in C. Similarly, we can show that G preserves I-injective maps
in C, and so G preserves trivial fibrations in C. Hence (i, G, ϕ) is a Quillen
adjunction.
We turn to the case of pointed categories [7, p.4]. Let D∗ be the pointed
category associated with D, and let U : D∗ → D be the forgetful functor.
We denote by I+ and J+ the classes of those maps f : X → Y in D∗ suchthat U f : U X → U Y belongs to I and J, respectively. Then we have thefollowing. (Compare [7, 1.1.8], [7, 1.3.5], and [7, 2.1.21].)
Theorem 2.2. Let D be a cofibrantly (resp. finitely) generated model category, and let C be a coreflective subcategory satisfying the conditions of Theorem 2.1. Then the pointed category C∗ has a cofibrantly (resp. finitely) generated model structure, with generating cofibrations I+ and generating trivial
cofibrations J+ , such that the induced adjunction (i∗ , G∗ , ϕ∗ ) : C∗ → D∗ isa Quillen adjunction.
We also have the following Proposition.
Proposition 2.3. Suppose C and D satisfy the conditions of Theorem 2.1.
Suppose, further, that the coreflection arrow ǫ : GY → Y is a weak equivalence for any fibrant object Y in D. Then the adjunctions (i, G, ϕ) : C → Dand (i∗ , G∗ , ϕ∗ ) : C∗ → D∗ are Quillen equivalences.
Proof. Let X be a cofibrant object in C and Y a fibrant object in D. Letf : X → Y be a map in D. Then we have ϕf = Gf ◦ η : X ∼= GX → GY .
ϕfǫ

Since f coincides with the composite X −−
→ GY −→ Y and ǫ is a weakequivalence in D, ϕf is a weak equivalence in C if and only if f is a weakequivalence in D. It follows by [7, 1.3.17] that that the induced adjunction(i∗ , G∗ , ϕ∗ ) is a Quillen equivalence.
3. On a model structure of the category NGIn [12] we introduced the notion of numerically generated spaces whichturns out to be the same notion as ∆-generated spaces introduced by Jeff
Smith (cf. [3]) . Let X be a topological space. A subset U of X is numerically
open if for every continuous map P : V → X, where V is an open subset of

82

TADAYUKI HARAGUCHI

Euclidean space, P −1 (U ) is open in V . Similarly, U is numerically closed
if for every such map P , P −1 (U ) is closed in V . A space X is called anumerically generated space if every numerically open subset is open in X.
Let NG denote the full subcategory of Top consisting of numericallygenerated spaces. Then the category NG is cartesian closed [12, 4.6]. To
any X we can associate the numerically generated space topology, denotedνX, by letting U open in νX if and only if U is numerically open in X.
Therefore we have a functor ν : Top → NG which takes X to νX. Clearly,
the identity map νX → X is continuous. By the results of [7, §3] thefollowing holds.
Proposition 3.1. The functor ν : Top → NG is a right adjoint to theinclusion functor i : NG → Top, so that NG is a coreflective subcategoryof Top.
A continuous map f : X → Y between topological spaces is called a weakhomotopy equivalence in Top if it induces an isomorphism of homotopy
groupsf∗ : πn (X, x) → πn (Y, f (x))
for all n > 0 and x ∈ X. Let I be the set of boundary inclusions S n−1 →D n , n ≥ 0, J the set of inclusions D n × {0} → D n × I, and WTop the classof weak homotopy equivalences. The standard model structure on Top canbe described as follows.
Theorem 3.2 ([7, 2.4.19]). There is a finitely generated model structure onTop with I as the set of generating cofibraitons, J as the set of generating
trivial cofibrations, and WTop as the class of weak equivalences.
The category NG is complete and cocomplete by [12, 3.4]. A space X
is numerically generated if and only if νX = X holds. Thus the unit ofthe adjunction η : X → νX is a natural homeomorphism. Moreover, since
CW-complexes are numerically generated spaces by [12, 4.4], the classes I
and J are contained in NG. Let WNG be the class of maps f : X → Yin NG which is a weak equivalence in Top. Since the coreflection arrowνY → Y , given by the identity of Y ∈ Top, is a weak equivalence (cf. [12,5.4]), we have the following by Theorem 2.1 and Proposition 2.3.
Theorem 3.3. The category NG has a finitely generated model structurewith I as the set of generating cofibrations, J as the set of generating trivial cofibrations, and WNG as the class of weak equivalences. Moreover theadjunction (i, ν, ϕ) : NG → Top is a Quillen equivalence.
We turn to the case of pointed spaces. Let Top∗ be the category of pointed
topological spaces. By [7, 2.4.20], there is a finitely generated model structure on the category Top∗ , with generating cofibrations I+ and generating

COREFLECTIVE SUBCATEGORIES OF A MODEL CATEGORY

83

trivial cofibrations J+ . Then we have the following by Theorem 2.2 andProposition 2.3.
Corollary 3.4. There is a finitely generated model structure on the categoryNG∗ of pointed numerically generated spaces, with generating cofibrations
I+ and generating trivial cofibrations J+ . Moreover, the inclusion functori∗ : NG∗ → Top∗ is a Quilen equivalence.
Remark. (1) The argument of Theorem 3.3 can be applied to the subcategories K of k-spaces and T of compactly generated spaces. Similarly, theargument of Corollary 3.4 can be applied to the pointed categories K∗ andT∗ . Compare [2.4.28], [2.4.25], [2.4.26] of [7].
(2) Let Diff be the category of diffeological spaces (cf. [8]). In [12] weintroduced a pair of functors T : Diff → Top and D : Diff → Top, where
T is a left adjoint to D, and showed that the composite T D coincides with
ν : Top → NG. Thus NG can be embedded as a full subcategory into
Diff . It is natural to ask whether Diff has a model category structure withrespect to which the pair (T, D) gives a Quillen adjuntion between Top andDiff .
Let I be the unit interval, and let λ : R → I be the smashing function,that is, a smooth function such that λ(t) = 0 for t ≤ 0 while λ(t) = 1 fort ≥ 1. Let I˜ denote the unit interval equipped with the quotient diffeologyλ∗ (DR ), where DR is the standard diffeology of R. In [5] we introduce
a finitely generated model category structure on Diff with the boundaryinclusions ∂ I˜n−1 → I˜n as generating cofibrations, and with the inclusions∂ I˜n−1 × I˜ ∪ I˜n × {0} → I˜n × I˜ as generating trivial cofibrations. Its classof weak equivalences consists of those smooth maps f : X → Y inducing
an isomorphism f∗ : πn (X, x0 ) → πn (Y, f (x0 )) for every n ≥ 0 and x0 ∈X. Here, the homotopy set πn (X, x0 ) is defined to be the set of smoothhomotopy classes of smooth maps (I˜n , ∂ I˜n ) → (X, x0 ).
It is expected that with respect to the model structure on Diff describedabove, the pair (T, D) induces a Quillen adjunction between Top and Diff .
Acknowledgements. I would like to express my sincere gratitude to my supervisor Kazuhisa Shimakawa. He introduced me to the project of building
a homotopy theory on the category NG. This project was not completed
without him. He carefully read this paper, helped me with the English andcorrected many errors. In order that I might acquire a doctor’s degree, hehad supported me for a long time.
Next I would like to thank Referee. He proposed that there exists a generalframework from proofs of Theorem 3.3 and Corollary 3.4. As a result, wehave Theorem 2.1, Theorem 2.2, and Proposition 2.3.

84

TADAYUKI HARAGUCHI

References[1] C. Berger and I. Moerdijk. Axiomatic homotopy theory for operads. Comment. Math.
Helv. , 78(4):805-831, 2003.
[2] S. E. Crans. Quillen closed model structures for sheaves. J. Pure Appl. Algebra ,101(1):35-57, 1995.
[3] D. Dugger, Notes on Delta-generated spaces, available at
http://www.uoregon.edu/ddugger/delta.html[4] W. G. Dwyer and J. Spalinski. Homotopy theories and model categories, Handbookof Algebraic Topology, Elsevier, 1995, 73-126.
[5] T. Haraguchi and K. Shimakawa, A model structure on the category of diffeologicalspaces, arXiv:1311.5668v2[6] P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys andMonographs, 99, American Mathematical Society, 2003.
[7] M. Hovey, Model categories, Mathematical Surveys and Monographs, 63, American
Mathematical Society, Providence, RI, 1999.
[8] P. Iglesias-Zemmour, Diffeology, CNRS, Marseille, France, and The Hebrew Universityof Jerusalem, Israel.
[9] J. W. Milnor. Topology from the differentiable viewpoint. Princeton Landmarks.
[10] D. G. Quillen, Homotopical Algebra, SLNM 43, Springer, Berlin (1967).
[11] D. G. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205-295.
[12] K. Shimakawa, K. Yoshida, and T. Haraguchi, Homology and cohomology via enriched bifunctors, arXiv:1010.3336v1.
[13] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York (1966).
Tadayuki Haraguchi
Department of General Education
Oita National College of Technology
1666, Oazamaki, Oita-Shi, Oita, 870-0152, Japane-mail address: t-haraguchi@oita-ct.ac.jp(Received November 2, 2012 )(Revised January 14, 2013 )