Next, we introduce a sort of cofibrations on a diagram category that are a bit more complicated to define.
For the purpose of this thesis, it suffices to consider the particular case where the shape category is given by S =∆n for some n∈N0, and so we restrict our attention to this case. A more general Reedy theory for Cisinski cofibration categories, where S may be a so-called finite directed category, can be found in the work of Rădulescu-Banu [30, ch. 9]. However, the Reedy cofibrations defined here are slightly more general than those of [30, def. 9.2.2(1)(b)] as we donot require a Reedy cofibration to have a Reedy cofibrant source object.
(3.82) Definition (Reedy cofibration). We suppose given a category with cofibrationsC and ann∈N0. (a) A morphism of∆n-commutative diagrams i: X → Y in C is called aReedy cofibration ifX and Y are
pointwise cofibrant, ifi0:X0→Y0is a cofibration inC, and if(Xk−1, Xk, Yk−1, Yk)is a coreedian rectangle in C fork∈∆n\ {0}.
(b) A ∆n-commutative diagram X in C is said to be Reedy cofibrant if it is cofibrant with respect to {i∈MorC∆n|iis a Reedy cofibration}.
(3.83) Remark. We suppose given a category with cofibrations C and an n ∈ N0. An S-commutative dia-gramX inC is Reedy cofibrant if and only ifX0 is cofibrant andXk−1,k is a cofibration inC fork∈∆n\ {0}.
Proof. First, we suppose that X is Reedy cofibrant, that is, there exists an initial objectI in C∆n such that iniIX:I → X is a Reedy cofibration. In particular, X is pointwise cofibrant, and so X0 is cofibrant in C.
Moreover, fork∈∆n\ {0}, the morphism Ik−1,k= iniIIk−1
k is an isomorphism, and soXk−1,k is a cofibration as (Ik−1, Ik, Xk−1, Xk)is coreedian.
Ik−1 Ik
Xk−1
Xk−1 Xk
∼=
iniIk−1
Xk−1 iniIk
Xk
iniIkXk−1
Conversely, we suppose that X0 is cofibrant and that Xk−1,k is a cofibration fork ∈ ∆n\ {0}. Then X is pointwise cofibrant by induction. Moreover, it is Reedy cofibrant as we have an initial object I in C∆n given byIk =¡C fork∈∆n and byIk,l= 1¡C for allk, l∈∆n with k≤l.
(3.84) Remark. We suppose given a category with cofibrationsC and ann∈N0. Every Reedy cofibration of
∆n-commutative diagrams inCis a pointwise cofibration. In particular, every Reedy cofibrant∆n-commutative diagram inCis pointwise cofibrant.
Proof. This follows from remark (3.56)(a) by an induction onn.
(3.85) Remark. We suppose given a category with cofibrations C and an n ∈ N0. Every isomorphism of
∆n-commutative diagrams inCwith pointwise cofibrant source object is a Reedy cofibration.
Proof. We suppose given an isomorphism of∆n-commutative diagramsf:X →Y inCsuch thatX is pointwise cofibrant. Thenf is a pointwise cofibration by the isomorphism axiom for cofibrations forCptw∆n. So in particular, Y is pointwise cofibrant and f0: X0 → Y0 is a cofibration. Moreover, (Xk−1, Xk, Yk−1, Yk) is coreedian for k∈∆n\ {0} by remark (3.57)(a).
(3.86) Proposition. We suppose given a category with cofibrationsC and ann∈N0.
(a) We suppose given a morphismf: X→Y and a Reedy cofibration of∆n-commutative diagramsi:X →X0 in C such thatX,Y,X0 are pointwise cofibrant. Then there exists a pushout rectangle
X0 Y0
X Y
f0
f
i i0
in C∆n.
(b) We suppose given a pushout rectangle
X0 Y0
X Y
f0
f
i i0
in C∆n such thatX, Y, X0 are pointwise cofibrant and such that i: X → X0 is a Reedy cofibration of
∆n-commutative diagrams in C. Then i0: Y → Y0 is a Reedy cofibration of ∆n-commutative diagrams in C.
Proof.
(a) Every Reedy cofibration is a pointwise cofibration by remark (3.84), so a pushout rectangle exists asCptwS fulfils the pushout axiom for cofibrations.
(b) By remark (3.84), iis a pointwise cofibration, and so i0 is a pointwise cofibration by remark (3.25). So in particular,Y0 is pointwise cofibrant andi00:Y0→Y00 is a cofibration. Moreover, for k∈∆n\ {0}, the coreedianess of(Xk−1, Xk, Xk−10 , Xk0)implies the coreedianess of(Yk−1, Yk, Yk−10 , Yk0)by proposition (3.60), whence i0:Y →Y0 is a Reedy cofibration.
Xk−10 Yk−10
Xk−1 Yk−1
Xk0 Yk0
Xk Yk
fk−10
fk0 fk−1
ik−1 i0k−1
fk
ik i0k
(3.87) Proposition. Given a category with cofibrationsCand ann∈N0, thenC∆n becomes a category with cofibrations having
CofC∆n={i∈MorC∆n|i is a Reedy cofibration}.
Proof. We setC:={i∈MorC∆n |iis a Reedy cofibration}. In the following, we verify the axioms of a category with cofibrations.
The categoryC∆nhas an initial objectIgiven byIk=¡C fork∈∆nand byIk,l= 1¡C for allk, l∈∆nwithk≤l.
Moreover,I is Reedy cofibrant asI0=¡C is cofibrant andIk−1,k= 1¡C is a cofibration inC fork∈∆\{0}.
To show thatCis closed under composition, we suppose given Reedy cofibrations of∆n-commutative diagrams i:X →Y,j:Y →Z inC, so thatX, Y,Z are pointwise cofibrant,i0:X0→Y0,j0: Y0→Z0 are cofibrations in C, and (Xk−1, Xk, Yk−1, Yk), (Yk−1, Yk, Zk−1, Zk) are coreedian rectangles inC fork ∈∆n\ {0}. But then i0j0: X0→Z0is also a cofibration inCby the multiplicativity ofCofC, and(Xk−1, Xk, Zk−1, Zk)is a coreedian rectangle inC fork∈∆n\ {0} by proposition (3.59)(a). Henceij: X→Z is a Reedy cofibration.
Xk−1 Xk
Yk−1 Yk
Zk−1 Zk
ik−1 ik
jk−1 jk
Finally, the isomorphism axiom for cofibrations follows from remark (3.85), and the pushout axiom for cofibra-tions follows from proposition (3.86).
(3.88) Definition (Reedy structure). We suppose given ann∈N0.
(a) Given a category with cofibrationsC, we denote byCReedy∆n the category with cofibrations whose underlying category is C∆n and whose set of cofibrations is
CofCReedy∆n ={i∈MorC∆n|iis a Reedy cofibration}.
The structure of a category with cofibrations of CReedy∆n is called theReedy structure (of a category with cofibrations) onC∆n.
(b) Given a category with cofibrations and weak equivalences C, we denote by CReedy∆n the category with cofibrations and weak equivalences whose underlying category is C∆n, whose underlying structure of a category with cofibrations is the Reedy structure of a category with cofibrations on C∆n, and whose underlying structure of a category with weak equivalences is the pointwise structure of a category with weak equivalences onC∆n. The structure of a category with cofibrations and weak equivalences ofCReedy∆n is called theReedy structure (of a category with cofibrations and weak equivalences) onC∆n.
(3.89) Remark. We suppose given a category with cofibrations and weak equivalencesC and ann∈N0. IfC fulfils the incision axiom, then so doesCReedy∆n .
Proof. This follows from proposition (3.86)(b) and remark (3.81).
(3.90) Proposition. We suppose given a category with cofibrations and weak equivalencesCthat fulfils the fac-torisation axiom for cofibrations, and we suppose given a morphism of∆n-commutative diagramsf:X →Y for somen∈N0. Moreover, we suppose given a Reedy cofibration of∆m-commutative diagramsires:X|∆m →Y˜res
and a pointwise weak equivalence of ∆m-commutative diagrams wres: ˜Yres → Y|∆m in C for some m ∈ N0
with m ≤ n such that f|∆m = ireswres. Then there exist a Reedy cofibration of ∆n-commutative dia-grams i: X → Y˜ and a pointwise weak equivalence of ∆n-commutative diagrams w: ˜Y → Y in C such that ires=i|∆m,wres=w|∆m andf =iw.
X0 . . . Xm . . . Xn
Y˜res,0 . . . Y˜res,m . . . Y˜n
Y0 . . . Ym . . . Yn
f0 ires,0
fm ires,m
fn in
wres,0 ≈
wres,m≈
wn ≈
Proof. For m =n, there is nothing to show. For m = 0, n = 1, the assertion follows from the factorisation lemma (3.65)(a). Form, n∈N0 withm < narbitrary, the assertion follows by an induction onn−m.
(3.91) Corollary. We suppose given a category with cofibrations and weak equivalencesCand ann∈N0. IfC fulfils the factorisation axiom for cofibrations, then so doesCReedy∆n andCptw∆n.
Proof. This follows from proposition (3.90) and remark (3.84).
For the definition of a Cisinski cofibration category and of a Brown cofibration category, see definition (3.51)(a) and definition (3.52)(a).
(3.92) Theorem. We suppose given a Cisinski cofibration categoryC and an n∈N0. ThenCReedy∆n and Cptw∆n are Cisinski cofibration categories.
Proof. This follows from remark (3.70), remark (3.81), remark (3.89) and corollary (3.91).
(3.93) Corollary. We suppose given a Brown cofibration category C and an n ∈N0. Then Cptw∆n is a Brown cofibration category.
Proof. This follows from theorem (3.92) and remark (3.80).