Fonds Grothendieck, cote n° 105, pages 21–40

[Structures de modèles fermées, paires et triples de Quillen]

titre de l'éditeur, entre crochets : les feuillets n'en portent aucun. Le texte est en anglais, tel qu'il l'écrit ; dans ce lot, la pagination de sa main commence à 9.

p. 9 de l'auteur. Alternative presentation of theory, with possibly slightly less strong result — as presentation of homotopy as conditions on a W, so that there should exist a closed model structure on A, s.th. C = \text{mono}, W = the given ones, hence TC = C \cap W, F defined by ~~lift.~~ RLP with respect to TC = C \cap W (or by (TC)_*), TF (« Kan- » maps) defined by RLP with respect to C (TF = C_*).

The problems which remain (as we know always (TF)^* = (C_*)^* = C)

ⓐ F^* = \struck{A_{\text{any}}}\ ((TC)_*)^* = TC — and as F^* \subset (TF)^* = C, this amounts to: any monomorphism which has the LLP with respect to F is in W. (A saturation type property of W with respect to monomorphisms.)

ⓑ TF = W \cap F — in terms of the assumptions already made previously upon W, we know TF \subset W, i.e. TF \subset W \cap F, thus the problem is W \cap F \subset TF, i.e. any f \in W which has the RLP with respect to W \cap C is in TF i.e. is Kan-.

c) Factorization f = pi with either p \in TF, i \in C (but this is already known and independent of W), or p \in F, i \in TC and even i \in C \cap W — and this shouldn't be too hard in terms of 3) and 4) by standard arguments. Thus I'll admit c for the time being. « and even i \in C \cap W » est en interligne. Les conditions 3), 4) et, plus bas, 5) sur W ne sont pas posées dans ce lot.

ⓓ Moreover, ~~I'm~~ I'm interested in the following, quite important for me, dual to (b): If f \in F, X \to Y, and (Y' \to Y) \in W, then (X' \to X) \in W.

For handling ⓐ ⓑ ⓓ presumably I'll have to rely strongly upon factorization c). Thus for a), using factorization of f = pi, and using f \in F^* we get a section s of p s.th. sf = i, thus f is a retract of i \in W, and if we assume 5).

\begin{tikzcd} & X' \arrow[d, "p \in F"] \\ X \arrow[ur, "i \in W \cap C"] \arrow[r, "f"'] & Y \arrow[u, bend right=40, "s"'] \end{tikzcd}

le petit diagramme est dans la marge inférieure, à gauche de la dernière ligne. La lettre après « Thus for » est tracée comme le c) qui précède ; mais l'argument (un monomorphisme ayant la LLP par rapport à F est rétracte de i \in W \cap C) est celui qui répond à ⓐ.

p. 10 de l'auteur. A category, if \Phi \subset \mathrm{Fl}(A), let \Phi^* = \{ u \in \mathrm{Fl}(A) \mid u \text{ has the LLP with respect to } \Phi \}, \Phi_* = \{ u \in \mathrm{Fl}(A) \mid u \text{ has the RLP with respect to } \Phi \} . \Phi is called left saturated if \Phi = \Psi^* for some \Psi, i.e. \Phi = (\Phi_*)^*; \Psi right saturated if ~~it is~~ \Psi = \Phi_* for some \Phi i.e. \Psi = (\Psi^*)_*. Left saturation, right saturation, \Phi \subset (\Phi_*)^*, \Psi \subset (\Psi^*)_*, associations \Phi \mapsto \Phi_*, \Psi \mapsto \Psi^* are order reversing bijections between left saturated and right saturated subsets.

Right saturation \Rightarrow stability by composition, base ch., retracts ; dually for left saturation. note écrite en oblique dans la marge gauche, en regard des lignes précédentes.

The factorization property for a pair (\Phi, \Psi) of mutually orthogonal subsets of \mathrm{Fl}(A) (\Phi \subset \Psi^* i.e. \Psi \subset \Phi_*), is the property ~~that~~ \forall f \in \mathrm{Fl}(A) \quad \exists \text{ factorization } f = pi, \text{ with } i \in \Phi,\ p \in \Psi . ~~Let be the left and right saturations respectively~~ ~~Prop.~~ ~~Let~~ Proposition (for orthogonal pairs (\Phi, \Psi)). Assume factorization property holds. Then \Phi = \Psi^* iff \Phi stable by direct factors. Dually \Psi = \Phi_* iff \Psi stable by retractions. Hence (\Phi, \Psi) are dual pair of subsets of \mathrm{Fl}(A) (localizers) iff ~~and~~ ~~(they are orthogonal)~~ both \Phi and \Psi are stable by direct factors.

Quillen pair (\Phi, \Psi): dual pair + factorization \Leftrightarrow orthogonal pair + factorization + stability of \Phi, \Psi (by direct factors).

Quillen triple (C, F, W) ~~Now let~~ with

a) W weakly saturated (isos in W, ~~and~~ and the and stable by direct factors) (M5)

b) (C, F \cap W \overset{\text{def}}{=} TF) and (TC \overset{\text{def}}{=} C \cap W, F) are Quillen pairs i.e. (M1) (M2) and stability of C, F, TC, TF ~~under~~ under direct factors (it is enough for C, F, provided it holds for W). \begin{array}{ccc} C & TF & \\ \cup & \cap & W \\ TC & F & \end{array} it would not be necessary here to assume the direct factor condition on W, if we assume b) cette insertion est reliée par une flèche au a).

(hence f \in W \Longleftrightarrow f = pi, i \in TC, p{} \in TF)

! This implies W is strongly saturated i.e. if \gamma_W : A \to W^{-1}A, f \in W \Longleftrightarrow \gamma_W(f) iso. true at any rate if in A finite direct limits and finite inverse limits exist (M0)

feuillet de brouillon sans numéro de l'auteur, barré de deux longues diagonales au crayon ; on le transcrit tel qu'il se lit. Y is Kan. X \to Y fibration, Kan.

\underline{\mathrm{Hom}}(I, Y) \to Y \times Y is a Kan fibration (if Y is Kan); is it a Serre fibration? a \to Y \times Y, \quad \alpha, \beta \in Y_a, \quad \underline{\mathrm{Hom}}_{\alpha,\beta}(I_a, Y_a), \quad \underline{\mathrm{Hom}}_{\alpha_b, \beta_b}(I_b, Y_b)

a) \mathrm{Iso}_M \subset W

b) If among u, v, uv two are in W, so is the third

c) W stable by retraction ~~(stronger than )~~

\Rightarrow mild saturation. But does it imply saturation, namely fg \in W, gf \in W \Longrightarrow f, g \in W??

A \underset{g}{\overset{f}{\rightleftarrows}} B fg \in W, gf \in W \Longrightarrow f, g \in W?

fg \in W_A, gf = \mathrm{id}_A \Longrightarrow f, g \in W

\begin{tikzcd} A \arrow[r, "f"] \arrow[d, bend left=30, "i"] & B \arrow[d, bend left=30, "j"] \\ A' \arrow[r, "f'"'] \arrow[u, bend left=30, "p"] & B' \arrow[u, bend left=30, "q"] \end{tikzcd}

f'i = jf, fp = qf', pi = \mathrm{id}_A, qj = \mathrm{id}_B, f' \in W \overset{?}{\Longrightarrow} f \in W.

(i) gf = \mathrm{id}_A, fg \in W \Rightarrow f, g \in W

(ii) gf \in W, fg \in W \Rightarrow f, g \in W

(iii) W stable ~~by factors~~ direct factors

(ii) \Rightarrow (i) \Leftarrow (iii)

autour de ces lignes, des esquisses : les carrés A \to A, A \xrightarrow{f} B, B \xrightarrow{fg} B exhibant f comme rétracte de fg ; un encadré « a) b) c_1) (c_2) (c_3) » ; en bas à gauche, A \rightleftarrows B avec « W », « {}^{h}W », gf \sim_h \mathrm{id}_A donc gf \in W, fg \sim_h \mathrm{id}_B donc fg \in W, et un triangle X \leftarrow I \times X ; en bas à droite, un schéma X \to X, X \wedge X qu'on ne restitue pas.

p. « 10 bis » de l'auteur. Le listing sur lequel il écrit porte imprimé dans sa marge « 06/09/82-09:14:22 » et « CFT 1.09 (05/10/82) ». X \xrightarrow{f} Y, X, Y \in M_{c,f}, f \in F.

Then f \in TF \Longleftrightarrow retract property \Longleftrightarrow i.e.

\begin{array}{cc} C & F \cap W = TF \\ \cup & \cap \\ TC = C \cap W & F \end{array} ⓐ \begin{cases} C = (TF)^* \\ F = (TC)_* \end{cases} ⓑ W = \bigl\{ pi \bigm| p \in \underbrace{C_*}_{\widetilde{TF}},\; i \in \underbrace{F^*}_{\widetilde{TC}} \bigr\}

\widetilde{TF} \subset F, \widetilde{TC} \subset C (because F, C trivially saturated)

~~TF = \widetilde{TF}, TC = \widetilde{TC} \} because TF = F \cap W, TC = C \cap W are stable under direct factors [provided we know W is stable]~~

we only have to ~~prove~~ express, to get of , that \boxed{\widetilde{TF} \subset W,\ \widetilde{TC} \subset W} \quad \text{and this is equivalent with} \quad \boxed{\widetilde{TF} = TF,\ \widetilde{TC} = TC} i.e. TF, TC saturated i.e. (C, TF), (TC, F) Quillen pairs.

Equivalent conditions on a model category (in Quillen's sense) ~~Closed model~~

(i) satisfies ⓐ and ⓑ, i.e. closed model category in Quillen sense

\Updownarrow (ii) (C, TF = F \cap W) and (TC = C \cap W, F) are Quillen pairs

(iii) C, F, W are stable under direct factors.

But I've to check that (i) (ii) (which are equivalent) does imply that W is stable under direct factors. But Quillen proves it even implies W is strongly saturated (i.e. f \in W iff \gamma(f) iso) — at least under the assumption that finite direct and inverse limits exist in M.

for any model category

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}] & & X_2 \arrow[r, "C \text{ or } F", "g"'] & Y_2 \arrow[dr, "TF"] & & \\ \varnothing \arrow[r, "C"] & X_1 \arrow[ur, "TC"] \arrow[dr, "TF"'] & & & Y_1 \arrow[r, "F"] & e \\ & & X \arrow[r, "f"'] & Y \arrow[ur, "TC"'] & & \end{tikzcd}

deux flèches en pointillé, marquées l'une « \varepsilon », l'autre « F », vont de X_1 et de X_2 vers Y_1 ; on ne les restitue pas. L'objet de départ est tracé \varphi : on le lit comme l'objet initial.

\gamma(f) iso \Longleftrightarrow \gamma(g) iso; f \in W \Longleftrightarrow g \in W.

W strongly saturated \Longleftrightarrow for f : X \to Y, X, Y \in M_{cf}, f \in C, \gamma(f) iso \Rightarrow f \in W \Longleftrightarrow for f : X \to Y, X, Y \in M_{cf}, f \in F, \gamma(f) iso \Rightarrow f \in W.

p. 11 de l'auteur. To construct any Quillen triple, we may proceed as follows: start with a localizer, C \subset \mathrm{Fl}(A), ⓐ C stable by direct factors, hence a pair (C, C_* = TF) — we must check ⓑ factorization condition for the pair (C, C_*). ~~Hence~~ Then (C, TF) is a Quillen pair. Let next ⓒ \boxed{W \supset TF} any localizer which is ⓓ mildly saturated and stable under direct factor. We now get TC = C \cap W \subset C, which is stable under direct factor. Take F = (TC)_* (\supset TF = C_*), we must check ⓔ factorization property for the pair (TC, F) — thus we get that (TC, F) is a Quillen pair. Lastly, we must still check that TF = F \cap W, and by c) we know already TF \subset F \cap W, we have to check still that F \cap W \subset TF. But using a factorization f = pi, p \in TF, i \in C, as f, p \in W we get i \in W hence i \in W \cap C = TC, hence there exists s : X' \to X s.th. si = \mathrm{id}_X (retraction of X' upon X) and fs = p, hence f a retraction of p, hence is in TF. OK.

\begin{tikzcd} X \arrow[r, "="] \arrow[d, "i"'] & X \arrow[d, "f \in F"] \\ X' \arrow[r, "p"'] \arrow[ur, dashed, "s"] & Y \end{tikzcd}

à côté, un second schéma : X' \rightleftarrows X par s et i, au-dessus de Y \rightleftarrows Y, avec p et f verticales — f rétracte de p. Le i du premier carré est marqué « \in TC ».

Thus

Proposition In order for a pair (C, W) to correspond to a Quillen triple (C, F, W) fibrations, it is n.s. that the following conditions be satisfied:

1)C, W stable by direct factors, W mildly saturated 2)Factorization conditions for (C, C_*) and for (C \cap W, (C \cap W)_*). 3)W \supset TF \overset{\text{def}}{=} C_*

We have to develop mainly a criterion to get factorization. [In the case of interest to us, we get 1) free, 3) almost free ( comes in the « test »-assumption though), and 2) causing a little technical problem.] I am however factorization conditions, interested in the in

p. 12 de l'auteur. following properties (characterizing a strict Quillen triple)

4) ~~A any cobase change of~~ W stable by base change by any g \in F, and by cobase change by any g \in C.

NB From my point of view, W is given beforehand in a rather tangible way, by a homological criterion, which turns out very handy. In Kan-Quillen's point of view, W was a little more hidden maybe in terms of (say) ss structures, they get to it via the pair (C, TF) and F \supset TF the Kan fibrations, which is a somewhat ad-hoc construction in terms of a a-priori generating set (TC)_0 — corresponding roughly to expressing the Kan lifting property. What remains to be understood, in the context of general test category, is the relationship between this and the direct description TC = C \cap W, i.e. why this is just the saturation of (TC)_0. Maybe I should have a look at Gabriel-Zisman and their treatment of « anodyne extensions ».

Factorization

titre de sa main, souligné, écrit en oblique dans la marge supérieure gauche du feuillet 33.

p. 13 de l'auteur. M a category, \Phi_0 \subset \mathrm{Fl}(M), \Psi = (\Phi_0)_*, \Phi_0 \subset \Phi \subset \overline{\Phi}_0 = \Psi^*, we want to deduce conditions on \Phi_0, \Phi insuring that the pair (\Phi, \Psi) satisfies factorization conditions, i.e. \forall f \in \mathrm{Fl}(M), \exists factorization f = pi, i \in \Phi, p \in \Psi. NB. we don't use \Phi \subset \uncertain{\overline{\Phi}_0} un astérisque est tracé au-dessus de \Phi_0 dans \Psi = (\Phi_0)_* ; sa portée n'est pas claire. L'insertion « NB. we don't use … » est au-dessus de la ligne.

the Problem which ; give the result p. 16^* note oblique dans la marge gauche, sous le titre ; « p. 16 » est sa pagination et renvoie au feuillet 39, qui énonce le théorème.

The idea is to take (for given f) ~~all~~ commutative diagrams

\begin{tikzcd} A \arrow[r, "u"] \arrow[d, "g"'] & X \arrow[d, "f"] \\ B \arrow[r, "v"'] & Y \end{tikzcd}

(D), with g \in \Phi_0, giving rise to

\begin{tikzcd} A \arrow[r, "u"] \arrow[d, "\alpha"'] & X \arrow[d, "i_D"] \arrow[dd, bend left=50, "f"] \\ B \arrow[r] & X_D = X \sqcup_A B \arrow[d, "p_D"] \\ & Y \end{tikzcd}

le carré supérieur est marqué « coc. » (cocartésien).

and to define \struck{\Sigma_1(f)}\; X_1(f) = \struck{\varinjlim}\; \coprod_{\substack{(X) \\ \text{all } D}} X_D \qquad \text{(amalgamated sum under } X) so that we get factorization

\begin{tikzcd} X \arrow[r, "i_1(f)"] \arrow[d, "f"'] & X_1(f) \arrow[dl, "\Sigma_1(f)"] \\ Y & \end{tikzcd}

with (hopefully) i_1(f) \in \Phi. le but de i_1(f) est d'abord écrit \Sigma_1(f), surchargé.

For this first step, we need

a) \Phi_0 is small (we'll see later how to relax this condition)

b) Amalgamated sums indexed by I with \mathrm{card}\, I \leqslant \mathrm{card}\, \Phi_0 or rather, \mathrm{card}\, \Phi_0 \times c, where c is \sup \mathrm{card}(\mathrm{Hom}(A, X) \times \mathrm{Hom}(B, Y)) exist in M, ~~and~~

c) \Phi stable by compositions, and by cobase extensions, and by filtering direct limits indexed by a set of cardinality \leqslant \mathrm{card}\, \Phi_0 rather \leqslant c for \alpha : A \to B in \Phi_0. insertions et renvois très serrés en bout de lignes b) et c) ; leur place exacte est incertaine.

do with the

We then are making an transfinite induction to define X_\alpha(f) and

\begin{tikzcd} X \arrow[r, "i_\alpha(f)"] \arrow[d, "f"'] & X_\alpha(f) \arrow[dl, "\Sigma_\alpha(f)"] \\ Y & \end{tikzcd}

i_\alpha(f) \in \Phi, with \alpha any ordinal < \alpha_0 (\alpha_0 ~~first~~ suitable ordinal () to be fixed later) ~~define later in terms of~~ : \begin{cases} \Sigma_{\alpha+1}(f) = \Sigma_1(\Sigma_\alpha(f)), \quad X_{\alpha+1}(f) = \text{source } \Sigma_{\alpha+1}(f), \\ i_{\alpha+1}(f) = \text{composition } X \xrightarrow{i_\alpha(f)} X_\alpha(f) \xrightarrow{i_1(\Sigma_\alpha(f))} X_{\alpha+1}(f) \\[1ex] \Sigma_\alpha(f) = \varinjlim_{\alpha' < \alpha} \Sigma_{\alpha'}(f) \quad \text{if } \alpha \text{ is a limiting ordinal.} \end{cases} en regard de la première ligne, un schéma : f : X \to Y, \Sigma_\alpha(f) : X_\alpha(f) \to Y, \Sigma_{\alpha+1}(f) = \Sigma_1(\Sigma_\alpha(f)) : X_{\alpha+1}(f) \to Y.

NB We assume that \Phi is stable too by the filtering direct limits of ~~cardinality \leqslant~~ which occur in the second transfinite induction step.

Let \pi be a cardinal such that ~~all objects of \Phi_0~~ for any g \in \Phi_0, source and target of g are \pi-accessible — i.e. \mathrm{Hom}(A, -) : M \to \text{Sets} commutes to filtering direct limits which are ~~great wr~~ « large with resp. to \pi ». We may take \pi = \sup_{g \in \Phi_0} \pi_g, where

p. 14 de l'auteur. \pi_g is the smallest cardinal with source and target of g are \pi-accessible.

Example If the objects occurring as source and targets of g's in \Phi_0 are « of finite presentation », i.e. \mathrm{Hom}(A, -) commutes with any filtering direct limit, we get \pi = \uncertain{0}. In this case we stop the induction with \alpha_0 = first infinite ordinal, ~~the first~~ ~~we mean~~. In case the objects A, B are not of finite presentation (but they are accessible) — we take the smallest ordinal such that \mathrm{Hom}(A, ?) commutes with direct limits indexed by ~~ordi~~ ordered set ~~great~~ large w.r.t. \pi. Thus, let \alpha_0 be the first ordinal such the interval I_{\alpha_0} = \{ \text{all ordinals } \alpha \text{ with } \alpha < \alpha_0 \} be ~~great~~ larger w.r.t. \pi. We take the factorization

\begin{tikzcd} X \arrow[r, "i_{\alpha_0}(f) = i(f)"] \arrow[d] & X_{\alpha_0}(f) = \underline{X}(f) \arrow[dl, "\Sigma_{\alpha_0}(f) = p(f)"] \\ Y & \end{tikzcd}

le « \pi = 0 » est tracé d'un rond épais ; on ne décide pas s'il s'agit de 0, de \omega ou d'autre chose. Dans le diagramme, un \Sigma est surchargé en \underline{X}.

and we get that i(f) \in \Phi, p(f) \in \Psi, provided we make the following assumptions:

d) The ~~objects~~ source and target objects of the arrows g \in \Phi_0 are accessible (ok if M a topos etc. —) and, if \pi_0 is the cardinality of the smallest ordinal such that for any such A, \mathrm{Hom}(A, -) commutes to the direct limits of type I_{\alpha_0}, ~~we assume~~ and ~~\pi = \sup(\pi_0~~ \pi \geqslant \sup(\pi_0, \mathrm{card}\, \Phi_0) rather, c, we assume that for any filtering direct systems with ordered set I of cardinality \leqslant \pi, and transitions maps in \Phi, the direct limit exists and Z_i \to Z are equally in \Phi.

Then we get, not only factorization f = pi, with i \in \Phi, p \in \Psi, but even factorization \underline{X}(f) functorial with respect to variable f. les deux dernières lignes sont marquées d'un double trait en marge.

p. 15 de l'auteur. Assume

a) \Phi_0 is small, source and target of any g \in \Phi_0 are accessible

b) M stable by filtering direct limits, ~~and by amalgamated~~

\begin{tikzcd} A \arrow[r, "u"] \arrow[d, "\alpha"'] & X \arrow[d, "\alpha'"] \\ B \arrow[r, "u'"'] & X' \end{tikzcd}

and \alpha' \in \Phi. Also c') \Phi stable under compositions

ⓓ \Phi « stable by filtering direct limits ». It is enough to know it w.r.t. ordered sets I where sup of any two elements exists

Then (\Phi, \Psi) satisfy the factorization condition, and ~~more~~ even, there exists a functor \Sigma

\begin{tikzcd} \underline{\mathrm{Fl}}(M) \arrow[rr, "\Sigma"] \arrow[dr, "t"'] & & \underline{\mathrm{Fl}}(M) \arrow[dl, "t"] \\ & M & \end{tikzcd}

and a functorial morphism in [\mathrm{Cat}]_{/M} possibly « large » categories over M \struck{\ill{}}\ \mathrm{id}_{\underline{\mathrm{Fl}}(M)} \to \Sigma with i(f) \in \Phi, p(f) \in \Psi.

\begin{tikzcd} X \arrow[r, "i(f)"] \arrow[d] & \underline{X}(f) \arrow[dl, "p(f)"] \\ Y & \end{tikzcd}

Example Assume that M is stable by amalgamated sums. Take \Phi = \overline{\Phi}_0 = \Psi^*. Then c) ~~and d)~~ is trivially satisfied. What about d)? OK for compositions. To get ~~extension of~~ the dotted \varphi, we make Zorn induction on pairs (J \subset I,\ u_J : A_J \to X) which make the diagram

\begin{tikzcd} A_{i_0} \arrow[r] \arrow[d, "u_0"'] \arrow[rr, bend left=30] & A_J \arrow[r] \arrow[dl, "u_J"] & B \arrow[d] \\ X \arrow[rr, "f"'] & & Y \end{tikzcd}

commutative, with obvious order relation. ~~This is clearly~~

à gauche, le schéma qui pose le problème : A_{i_0} \to A_i \to B = \varinjlim A_i, u_0 : A_{i_0} \to X, v : B \to Y, f : X \to Y, et la flèche cherchée \varphi : B \to X en pointillé, marquée « induction ». La paire de la récurrence est écrite (J \subset I, u_J : \varinjlim_J A_j \to X), avec une surcharge. \mathrm{Hom}(B, X) \to \mathrm{Hom}(g_i, f) \quad \text{surjective}, \qquad \mathrm{Hom}(g, f) = \varprojlim \mathrm{Hom}(g_i, f)

p. 16 de l'auteur. Lemma Let \Phi \subset \mathrm{Fl}(M) be left Q-saturated (i.e. \Phi = \Psi^* for some \Psi). Then ~~\Phi is stable by compositions,~~

a) \Phi contains iso

b) \Phi stable by under composition and cobase changes (including )

c) \Phi stable by under ~~direct~~ arbitrary amalgamated sums

d) \Phi stable ~~by direct limits, indexed by any well ordered set i.e.~~ under ordinal direct systems, i.e. ~~if we have any directed ordered set~~ if we have any well ordered set I and ~~system~~ direct system (A_i)_{i \in I} with g_{ij} : A_i \to A_j in \Phi, and A_{i_0} = \varinjlim_{j < i_0} A_j if i_0 \in I is a limit, then \forall i_0 \in I, A_{i_0} \xrightarrow{g_{i_0}} B = \varinjlim A_i is in \Phi. NB a), b), d) \Rightarrow c)

~~The only~~

Thus we get,

Theorem Let M be a category, stable by small direct limits, and (weakly) accessible. Then

① Let \Phi_0 \subset \mathrm{Fl}(M) ^*, and \Phi_0 \subset \Phi \subset \overline{\Phi}_0 \overset{\text{def}}{=} \Psi^* (with \Psi = \Phi_{0*}). If \Phi ~~stable and~~ satisfies a) b) c) d)^*, then the (\Phi, \Psi) satisfies factorization condition. (Hence, if \Phi stable by direct factors, \Phi = \overline{\Phi}_0.)

② (In order that \Phi (some data \Phi_0, \Phi) be saturated, \Phi = \overline{\Phi}_0, it is n.s. that \Phi satisfy conditions a) b) c) d) and e) stability by direct factors. In this case, factorization holds for the Q-orthogonal pair (\Phi, \Psi).

We have to assume \Phi_0 small (). We may (which results from a) b) d)) if \Phi_0 . Alternatively a) b') c) d), where b') is stability under cobase changes (without composition). note oblique dans la marge gauche, en regard de ① et ② ; les astérisques renvoient à cette note.

NB. In 2), ~~besides~~ there was the preliminary assumption of existence of \Phi_0 \subset \Phi s.th. \Phi_0 and \Phi the same left Q-saturation, with \Phi_0 small. Next step is to try to get rid of this.