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First of all, since M SM_S belongs to a prefactorization system, it is closed under composites, pullbacks, and any intersections which exist. Therefore, if we define MMM SM' \coloneqq M \cap M_S, then MM' satisfies the hypotheses of Theorem \ref{ConstructingOFS}, and so we have an OFS (E,M)(E',M').

Moreover, it is useful to notice that E S=E_S=: this is an easy consequence of the fact that if STS\dashv T, then SabaTbSa\perp b\iff a\perp Tb, since fTuSfuf\perp Tu\iff Sf\perp u for each uhom(C)u\in\hom(C), so that SfS f is an isomorphism.

Now suppose given f:ABf\colon A\to B; we want to construct an (E S,M S)(E_S,M_S)-factorization. Let vv be the pullback of TSfT S f along the unit η B:BTSB\eta_B \colon B \to T S B. The naturality square for η\eta at ff shows that ff factors through vv, say f=vwf = v w.

A w P u TSA v TSf B η B TSB\begin{array}{ccccc} A & \\ &\overset{w}\searrow\\ && P &\overset{u}\to& T S A \\ && {}^v\downarrow && \downarrow^{T S f}\\ && B &\underset{\eta_B}\to& T S B \end{array}

Since TSfT S f is evidently in M S=( T(hom(C))) T(hom(C))M_S=({}^\perp T(\hom(C)))^\perp\supseteq T(\hom (C)), so is vv; thus it suffices to find an (E S,M S)(E_S,M_S)-factorization of ww.

Let w=ngw = n g be the (E,M)(E',M')-factorization of ww. Since MM SM' \subseteq M_S, it suffices to show that gE Sg\in E_S. Note also that since ww is a first factor of the unit η A\eta_A, by passing to adjuncts we find that SwS w is split monic: in the former diagram we have uw=η Au w=\eta_A, so that the adjunct ϵ SASuSnSg=1\epsilon_{S A} \cdot S u\cdot S n\cdot S g=1, hence also SgS g is a split monic. But TSgT S g is then also split monic, hence belongs to MM and thus also to MM' (since it obviously belong to M S=( T(hom(C))) T(hom(C))M_S=({}^\perp T(\hom(C)))^\perp\supseteq T(\hom (C))). Therefore, since gEg\in E', the naturality square for η\eta at gg contains a lift: there is an α:XTSA\alpha\colon X\to T S A such that in the diagram

A η A TSA g TSg X η X TSX\begin{array}{ccc} A & \overset{\eta_A}\to & T S A \\ {}^g\downarrow && \downarrow^{T S g} \\ X &\overset{\eta_X}\to& T S X \end{array}

αg=η A\alpha\cdot g=\eta_A and TSgα=η XT S g \cdot \alpha=\eta_X. Passing to adjuncts again, we find that SgS g is also split epic, since we can consider the diagram

SA Sη A STSA ϵ SA SA Sg aaa STSg Sg SX Sη X STSX ϵ SX SX\begin{array}{ccccc} S A & \overset{S\eta_A}\longrightarrow & S T S A &\overset{\epsilon_{SA}}\longrightarrow & S A\\ {}^{S g}\downarrow && \phantom{aaa}\downarrow_{S T S g} &&\downarrow^{S g}\\ S X &\underset{S\eta_X}\longrightarrow & S T S X &\underset{\epsilon_{SX}}\longrightarrow & S X \end{array}

and the commutativity

Sgϵ SASα=ϵ SXSTSgSα=ϵ SXSη X=1 S g \cdot \epsilon_{S A} \cdot S\alpha = \epsilon_{S X} S T S g \cdot S\alpha = \epsilon_{S X}\cdot S\eta_X = 1

Hence SgS g is an isomorphism; thus gE Sg\in E_S as desired. \blacksquare


?? Grothendieck inequality <>&lt;&gt; ***

  1. [,𝒜][\mathcal{I},\mathcal{A}] is really just the underlying category with hom-collections given by A 0(A,B)=V 0(I,𝒜(A,B))A_0(A,B)=V_0(I,\mathcal{A}(A,B)).
  2. 𝒜(,)\mathcal{A}(-,-) is the fully faithful two-variable hom-functor from A 0 op×A 0V 0A_0^{op}\times A_0\to V_0, with 𝒜(f,g)\mathcal{A}(f,g) defined as the composite 𝒜(B,C)l 1r 1I𝒜(B,C)Ifidg𝒜(C,D)𝒜(B,C)𝒜(A,B)𝒜( 𝒜) 2(A,D)\mathcal{A}(B,C)\stackrel{l^{-1}r^{-1}}{\to}I\otimes\mathcal{A}(B,C)\otimes I\stackrel{f\otimes id\otimes g}{\to}\mathcal{A}(C,D)\otimes\mathcal{A}(B,C)\otimes\mathcal{A}(A,B)\stackrel{(\circ^{\mathcal{A}})^2}\mathcal{A}(A,D) in V 0V_0
  3. [,F][\mathcal{I},F] is the functor from A 0A_0 to B 0B_0 underlying the enriched functor FF. This is defined by letting FfFf be the composite If𝒜(A,B)F A,B(FA,FB)I\stackrel{f}{\to}\mathcal{A}(A,B)\stackrel{F_{A,B}}\mathcal{B}(FA,FB) where F A,BF_{A,B} is the family of morphisms in V 0V_0 defining the enriched functor FF.
  4. The natural transformation F¯:catA(,)catB(F,F)\bar F\colon\cat A(-,-)\to\cat B(F-,F-) has for its components exactly the maps F A,BF_{A,B} above: i.e. F¯ A,B=F A,B\bar F_{A,B}=F_{A,B}. * *
    category: meta

Revised on April 20, 2014 09:13:30 by Fosco Loregian (79.1.178.57)