Cofibrant objects

In a category with cofibrations C, one has the notion of a cofibrant object with respect to the set of cofibra-tionsCofC, see definition (3.13). We abbreviate the terminology:

(3.19) Definition (cofibrant object). We suppose given a category with cofibrations C. The full subcate-goryCcof :=C_(CofC)-cof of C is called the full subcategory of cofibrant objects in C. An object in C that lies in Ccof is said to becofibrant, and a morphism inCcof is called amorphism of cofibrant objects in C.

Given a category with cofibrationsC, thereexists an initial object inCthat is cofibrant. Moreover, an objectX in Cis cofibrant if there existsan initial object I such that the unique morphismini^I_X:I→X is a cofibration in C. The following two remarks show the independence of the notion of cofibrancy from the considered initial object. Likewise, remark (3.25) shows the independence of the considered pushout in the pushout axiom for cofibrations.

(3.20) Remark. Every initial object in a category with cofibrations is cofibrant.

Proof. We suppose given a category with cofibrations C and an initial object I in C. There exists an initial objectI⁰ inC that is cofibrant, and so asini^I_I⁰:I⁰→I is an isomorphism inC, it is a cofibration inC. But this means thatI is cofibrant.

(3.21) Remark. We suppose given a category with cofibrationsC. An objectX inC is cofibrant if and only if for every initial objectI in C, the unique morphism ini^I_X:I→X is a cofibration inC.

Proof. We suppose given an object X in C. First, we suppose that X is cofibrant, that is, we suppose that there exists an initial objectI⁰ inCsuch thatini^I_X⁰:I⁰→X is a cofibration. Moreover, we letI be an arbitrary initial object inC. Then the unique morphismini^I_I0:I→I⁰ is an isomorphism and therefore a cofibration asI is cofibrant by remark (3.20). But then also ini^I_X = ini^I_I0ini^I_X⁰ is a cofibration as cofibrations are closed under composition. Conversely, ifini^I_X:I→X is a cofibration for every initial objectIinC, thenX is cofibrant since there exists an initial object inC.

If unambiguous, we will consider the full subcategory of cofibrant objects in a category with cofibrations, see definition (3.19) and definition (3.14)(a), as a category with cofibrations in the following way, without further comment.

(3.22) Remark. Given a category with cofibrationsC, the full subcategory of cofibrant objectsCcof becomes a category with cofibrations having

CofCcof= CofC ∩MorCcof.

Moreover,CofCcof is a multiplicative subset ofMorCcof.

(3.23) Remark. Given a category with cofibrationsC, then every cofibration inCwith cofibrant source object has a cofibrant target object.

Proof. We suppose given a cofibrationi:X →Y in Csuch thatX is cofibrant. TheniniX is a cofibration, and henceiniY = iniXiis a cofibration as cofibrations are closed under composition. ThusY is cofibrant.

(3.24) Corollary. Given a category with cofibrationsC, the full subcategory of cofibrant objectsCcofis closed under isomorphisms inC.

Proof. We suppose given an isomorphismf:X →Y inC such that X is cofibrant. Thenf is a cofibration by the isomorphism axiom for cofibrations, and henceY is cofibrant by remark (3.23).

(3.25) Remark. We suppose given a category with cofibrations Cand a pushout rectangle

X⁰ Y⁰

X Y

f⁰

f

i i⁰

inC such thatf:X →Y is a morphism andi:X →X⁰ is a cofibration inC_cof. Theni⁰ is a cofibration andY⁰ is cofibrant inC.

Proof. AsC is a category with cofibrations, there exists a pushout rectangle X⁰ Y˜⁰

X Y

f˜⁰

f

i ˜i⁰

in C such that˜i⁰: Y → Y˜⁰ is a cofibration. Then Y˜⁰ is cofibrant since Y is cofibrant and˜i⁰ is a cofibration.

Moreover, since(X, Y, X⁰,Y˜⁰)and(X, Y, X⁰, Y⁰)are pushout rectangles inC, the unique morphismg: ˜Y⁰ →Y⁰ withf⁰= ˜f⁰g andi⁰ = ˜i⁰gis an isomorphism. By the isomorphism axiom for cofibrations, it follows that gis a cofibration asY˜⁰ is cofibrant. In particular,i⁰ = ˜i⁰gis a cofibration as cofibrations are closed under composition, andY⁰ is cofibrant by remark (3.23).

Y⁰

X⁰ Y˜⁰

X Y

f˜⁰ f⁰

g ∼=

f

i ˜i⁰

i⁰

(3.26) Proposition (cf. [30, lem. 1.2.1(1)]). We suppose given a category with cofibrations C.

(a) The full subcategory of cofibrant objects Ccof has finite coproducts. Given n ∈ N⁰ and objects Xk

in Ccof for k ∈ [1, n], the coproduct `

k∈[1,n]Yk is a cofibration.

Proof.

(a) As ¡ is cofibrant, for cofibrant objects X1, X2 in C, there exists a pushout C of iniX₁ and iniX₂ by the pushout axiom for cofibrations. The embeddings emb1 and emb2 are cofibrations and C is cofibrant by remark (3.25). Moreover,C is a coproduct ofX₁ andX₂ inC.

The assertion follows by induction, using the closedness ofCofCcofunder composition and the isomorphism axiom for cofibrations.

Forn∈N0 arbitrary, the assertion follows by induction.

(3.27) Corollary. We suppose given a category with cofibrationsC, ann∈ N0 and morphisms ik:Xk →Y in Ccof fork ∈[1, n]. If (ik)_k∈[1,n] : `

k∈[1,n]X_k → Y is a cofibration in C, theni_k:X_k →Y is a cofibration inC for every k∈[1, n].

Proof. As X_k for k ∈ [1, n] is cofibrant, the embedding emb_k:X_k →`

j∈[1,n]X_j is a cofibration by proposi-tion (3.26)(a). So if(ⁱk)_k∈[1,n] is a cofibration, thenik = embk(ⁱj)_j∈[1,n] is a cofibration for everyk∈[1, n]by closedness under composition.

(3.28) Proposition. We suppose given a category with cofibrations C and morphisms i1: X1 → X, i2:X2→X,f:X2→Y inCcof. If ⁱ_i¹

Proof. We have

(3.29) Definition (cofibrancy axiom). A category with cofibrations C is said to fulfil thecofibrancy axiom if the following holds.

(Cof) Cofibrancy axiom. Every object inC is cofibrant.

3 Categories with cofibrations and weak equivalences

In this section, we combine the notion of a category with weak equivalences from section 1 with that of a category with cofibrations from section 2 and introduce the notion of category with cofibrations and weak equivalences, see definition (3.30)(a). The two underlying structures given by the cofibrations on the one hand, and by the weak equivalences on the other hand, are completely independent so far; there are no axioms that describe the interplay between cofibrations and weak equivalences. This will be done in the next section 4, where we present some properties such a category with cofibrations and weak equivalences may fulfil.

Definition of a category with cofibrations and weak equivalences

For the definition of a category with cofibrations and of a morphism of categories with cofibrations, see defini-tion (3.14). For the definidefini-tion of a category with weak equivalences and of a morphism of categories with weak equivalences, see definition (3.1).

(3.30) Definition (category with cofibrations and weak equivalences).

(a) A category with cofibrations and weak equivalences consists of a category C together with subsets C, W ⊆ MorC such that C becomes a category with cofibrations having CofC = C and a category with weak equivalences having WeC=W.

(b) We suppose given categories with cofibrations and weak equivalencesCandD. Amorphism of categories with cofibrations and weak equivalences fromCtoDis a functorF:C → Dthat is a morphism of categories with cofibrations and a morphism of categories with weak equivalences.

As for categories with weak equivalences, cf. definition (3.10), we can define a zero-pointed variant, which will become important in chapter V.