LetC be a category containing a collection of morphisms{fα:Cα →Dα}, and letg :X →Z be another morphism in C. Under some mild hypotheses, Quillen’s small object argument can be used to produce a factorization
X g
0
→Y g
00
→Z
whereg0is “built from” the morphismsfα, andg00has the right lifting property with respect to the morphisms fα(see§T.A.1.2 for a detailed discussion). The small object argument was originally used by Grothendieck to prove that every Grothendieck abelian category has enough injective objects (see [25] or Corollary A.1.3.4.7).
It is now a basic tool in the theory of model categories.
Our goal in this section is to carry out an∞-categorical version of the small object argument (Proposition 1.4.7). We begin by introducing some terminology.
Definition 1.4.1. LetCbe an∞-category. Letf :C→D andg:X →Y be morphisms inC. We will say thatg has theright lifting propertywith respect tof if every commutative diagram
C //
f
X
g
D //Y can be extended to a 3-simplex ofC, as depicted by the diagram
C //
f
X
g
D
>>//Y.
In this case, we will also say thatf has theleft lifting propertywith respect tog.
More generally, if S is any set of morphisms inC, we will say that a morphism g has the right lifting propertywith respect toS if it has the right lifting property with respect to every morphism inS, and that a morphismf has theleft lifting propertywith respect toS iff has the left lifting property with respect to every morphism inS.
Definition 1.4.2. LetCbe an∞-category and let S be a collection of morphisms inC. We will say that a morphism f in C is a transfinite pushout of morphisms in S if there exists an ordinal αand a diagram F : N[α]→C(here [α] denotes the linearly ordered set of ordinals{β :β ≤α}) with the following properties:
(1) For every nonzero limit ordinalλ≤α, the restrictionF|N[λ] is a colimit diagram inC. (2) For every ordinalβ < α, the morphismF(β)→F(β+ 1) is a pushout of a morphism inS.
(3) The morphismF(0)→F(α) coincides withf.
Remark 1.4.3. LetC be an∞-category, and let S andT be collections of morphisms inC. Suppose that every morphism belonging toT is a transfinite pushout of morphisms inS. Iff is a transfinite pushout of morphisms inT, thenf is a transfinite pushout of morphisms inS.
Definition 1.4.4. LetCbe an∞-category and letS be a collection of morphisms inC. We will say thatS isweakly saturatedif it has the following properties:
(1) Iff is a morphism inCwhich is a transfinite pushout of morphisms inS, then f ∈S.
(2) The setS is closed under retracts. In other words, if we are given a commutative diagram C
f //C0
f0
//C
f
D //D0 //D
in which both horizontal compositions are the identity andf0 belongs toS, then so does f.
Remark 1.4.5. If C is the nerve of an ordinary category (which admits small colimits), then Definition 1.4.4 reduces to Definition T.A.1.2.2.
Remark 1.4.6. Let S be a weakly saturated collection of morphisms in an∞-category C. Any identity map inC can be written as a transfinite composition of morphisms in S (take α= 0 in Definition 1.4.2).
Condition (2) of Definition 1.4.4 guarantees that the class of morphisms is stable under equivalence; it follows that every equivalence inCbelongs toS. Condition (1) of Definition 1.4.4 also implies thatSis closed under composition (takeα= 2 in Definition 1.4.2).
We can now formulate our main result, which we will prove at the end of this section.
Proposition 1.4.7 (Small Object Argument). Let C be a presentable ∞-category and let S be a small collection of morphisms in C. Then every morphismf :X→Z admits a factorizaton
X f
0
→Y f
00
→Z
wheref0 is a transfinite pushout of morphisms inS andf00 has the right lifting property with respect toS.
Warning 1.4.8. In contrast with the ordinary categorical setting (see Proposition T.A.1.2.5), the factor-ization
of Proposition 1.4.7 cannot generally be chosen to depend functorially onf.
To apply Proposition 1.4.7, the following observation is often useful:
Proposition 1.4.9. Let C be an ∞-category and let T be a collection of morphisms in C. Let S denote the collection of all morphisms in Cwhich have the left lifting property with respect toT. Then S is weakly saturated.
Proof. Since the intersection of a collection of weakly saturated collections is weakly saturated, it will suffice to treat the case where T consists of a single morphism g : X → Y. Note that a morphism f : C → D has the left lifting property with respect tog if and only if, for every lifting of Y to Cf /, the induced map θf : Cf / /Y → CC/ /Y is surjective on objects which lie over g ∈ C/Y. Sinceθf is a left fibration, it is a categorical fibration; it therefore suffices to show that object of CC/ /Y which lies overg is in the essential image ofθf. We begin by showing thatSis stable under pushouts. Suppose we are given a pushout diagram σ: of g to Cλ/ /Y lies in the essential image of θ, which follows from our assumption that every lifting of g to CC0/ /Y lies in the essential image ofθf0.
We now verify condition (1) of Definition 1.4.4. Fix an ordinary αand a diagramF : [α]→Csatisfying the hypotheses of Definition 1.4.2, and letf :F(0)→F(α) be the induced map. Choose a lifting ofY toCf / remains to treat the case of a successor ordinal: letβ < αand assume thatXβ has been defined; we wish to show that the vertexXβ lies in the image of the mapθβ:CFβ+1/ /Y →CFβ/ /Y. Letu:F(β)→F(β+ 1) be the morphism determined byF, so thatθβ is equivalent to the mapθu. Since the image ofXβ inCF(β)/ /Y
lies overg, the existence of the desired lifting follows from our assumption thatu∈S.
We now verify (2). Consider a diagram σ: ∆2×∆1→C, given by
Assume thatf0 ∈S; we wish to provef ∈S. Choose a lifting ofY toCf /, and liftY further toCσ/ (here we identifyf withσ|({2} ×∆1). LetX be a lifting ofg toCC/ /Y; we wish to show thatX lies in the image of θf. We can lift X further to an objectXe ∈Cσ0/ /Y, where σ0 =σ|(∆2× {0}). Letσ0 =σ|(∆1×∆1).
The forgetful functor θ:Cσ0/ /Y →Cλ/ /Y is equivalent toθf0, so that the image ofXe in Cλ/ /Y lies in the image ofθ. It follows immediately thatX lies in the image ofθf.
Corollary 1.4.10. Let Cbe a presentable ∞-category, let S be a small collection of morphisms of C, letT be the collection of all morphisms in Cwhich have the right lifting property with respect to every morphism in S, and letS∨ be the collection of all morphisms in C which have the left lifting property with respect to every morphism in T. Then S∨ is the smallest weakly saturated collection of morphisms which containsS.
Proof. Proposition 1.4.9 implies thatS∨is weakly saturated, and it is obvious thatS∨containsS. Suppose that S is any weakly saturated collection of morphisms which containsS; we will show that S∨ ⊆S. Let f :X →Z be a morphism in S∨, and choose a factorization
which shows thatf is a retract off0 and therefore belongs toS as desired.
Recall that ifCis an∞-category which admits finite limits and colimits, then every simplicial objectX• of C determines latching and matching objects Ln(X•), Mn(X•) for n ≥0 (see Remark T.A.2.9.16). The following result will play an important role in our proof of Theorem 1.3.12:
Corollary 1.4.11. Let C be a presentable ∞-category and let S be a small collection of morphisms in C. Let Y be any object of C, and letφ:C/Y →C be the forgetful functor. Then there exists a simplicial object X• ofC/Y with the following properties:
(1) For each n≥0, let un :Ln(X•)→Xn be the canonical map. Then φ(un) is a transfinite pushout of morphisms inS.
(2) For each n≥0, letvn :Xn →Mn(X•)be the canonical map inC/Y. Thenφ(vn) has the right lifting property with respect to every morphism inS.
Proof. We construct X• as the union of a compatible family of diagramsX•(n) : N(∆≤n)op →C/Y, which we construct by induction onn. The casen=−1 is trivial (since∆≤−1 is empty). Assume thatn≥0 and that X•(n−1) has been constructed, so that the matching and latching objects Ln(X), Mn(X) are defined and we have a mapt:Ln(X)→Mn(X). Using Proposition T.A.2.9.14, we see that it suffices to construct
in C/Y. Since the map C/Y → C is a right fibration, this is equivalent to the problem of producing a commutative diagram
in the∞-categoryC. Proposition 1.4.7 guarantees that we are able to make these choices in such a way that (1) and (2) are satisfied.
Remark 1.4.12. In the situation of Corollary 1.4.11, let ∅ denote the initial object of C. Then for each n≥0, the canonical map w:∅ →φ(Xn) is a transfinite pushout of morphisms in S. To prove this, we let P denote the full subcategory of∆[n]/ spanned by the surjective maps [n] → [m]; we will regard P as a partially ordered set. For each upward-closed subset P0 ⊆P, we letZ(P0) denote a colimit of the induced diagram
N(P0)op−→N(∆)op X−→• C/Y
−→φ C.
ThenZ(∅)' ∅ andZ(P)'φ(Xn). It will therefore suffice to show that if P1 ⊆P is obtained from P0 by adjoining a new element given by α: [n]→[m], then the induced map θ: Z(P0)→Z(P1) is a transfinite pushout of morphisms inS. This follows from assertion (1) of Corollary 1.4.11, sinceθ is a pushout of the mapφ(un) :φ(Lm(X•))→φ(Xm).
Proof of Proposition 1.4.7. Let S ={gi :Ci →Di}i∈I. Choose a regular cardinal κsuch that each of the objects Ci is κ-compact. We construct a diagram F : N[κ] → C/Z as the union of maps {Fα : N[α] → C/Z}α≤κ; here [α] denotes the linearly ordered set of ordinals{β : β ≤α}. The construction proceeds by induction: we let F0 be the morphism f : X → Z, and for a nonzero limit ordinal λ ≤ κwe let Fλ be a colimit of the diagram obtained by amalgamating themaps {Fα}α<λ. Assume that α < κ and that Fα
been constructed. ThenFα(α) corresponds to a mapX0→Z. LetT(α) be a set of representantives for all equivalence classes of diagrams equivalence classes of diagramsσt:
Ct //
gt
X0
Dt //Z,
wheregt is a morphism belonging toS. Choose a pushout diagram
`
inC/Z. We regardX00as an object ofCX0/ /Z. Since the map (C/Z)Fα/→CX0/ /Z is a trivial Kan fibration, we can liftX0 to an object of (C/Z)Fα/, which determines the desired mapFα+1.
For eachα≤κ, letfα:Yα→Z be the image F(α)∈C/Z. LetY =Yκ andf00=fκ. We claim thatf00 has the right lifting property with respect to every morphism inS. In other words, we wish to show that for eachi∈I and every mapDi→Z, the induced map
MapC
/Z(Di, Y)→MapC
/Z(Ci, Y)
is surjective on connected components. Choose a pointη∈MapC/Z(Ci, Y). SinceCiisκ-compact, the space MapC
/Z(Ci, Y) can be realized as the filtered colimit of mapping spaces lim
−→αMapC showing thatf0 is a transfinite pushout of morphisms inS. Using Remark 1.4.3, we are reduced to showing that for eachα < κ, the mapYα→Yα+1 is a transfinite pushout of morphisms inS. To prove this, choose a well-ordering of T(α) having order type β. For γ < β, let tγ denote the corresponding element ofT(α).
We define a functorG: N[β]→Cso that, for each β0≤β, we have a pushout diagram
It is easy to see that Gsatisfies the conditions of Definition 1.4.2 and therefore exhibits Yα → Yα+1 as a transfinite pushout of morphisms inS.