Cote n° 105 · batch 2 · pages 21–40 · Transcription · [Champs (stacks) 2] : notes manuscrites (s.d.).
Datation de l’inventaire : [à partir de 1982]
Édition de démonstration

[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.

21p. 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 ⓐ.

24p. 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)

26feuillet 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.

27p. « 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 \(\Rightarrow f \in W\).

29p. 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. 1)\(C\), \(W\) stable by direct factors, \(W\) mildly saturated
2. 2)Factorization conditions for \((C, C_*)\) and for \((C \cap W, (C \cap W)_*)\).
3. 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

31p. 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.

33p. 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

35p. 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.

37p. 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 […]

ⓒ Any [For any] amalgamated diagram \(A \xrightarrow{u} X\), \(\alpha : A \to B\), with \(\alpha \in \Phi_0\), the amalgamated sum exists

\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. \[ \mathrm{Hom}(B, X) \to \mathrm{Hom}(g_i, f) \quad \text{surjective}, \qquad \mathrm{Hom}(g, f) = \varprojlim \mathrm{Hom}(g_i, f) \]

39p. 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.