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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 7th 2010
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexBefore asking this question, we must recall some notions: ------- Fix an excellent (therefore symmetric monoidal closed, combinatorial, and all monomorphisms are cofibrations (there's more, but we don't need it now)) model category $S$ (note: $S$ behaves in almost every way like the category of simplicial sets with the cartesian monoidal structure. If you're not familiar with the definition of excellent, pretend that $S$ is the category of simplicial sets). We say that an $S$-enriched category $C$ is $A$-filtered, or equipped with an $A$-filtration, for a fixed poset $A$ if there exists a function $r:Ob(C)\to A$ such that if $r(X)\nleq r(Y)$, $Map_C(X,Y)=\emptyset$ where $\emptyset$ is the initial object of $S$. Any $A$-filtered $S$-category gives rise to a functor $A\to Cat_S$ as follows: Let $C_{\leq a}$ be the full subcategory of $C$ spanned by the set of objects $X$ such that $r(X)\leq a$. This gives a filtered diagram of subcategories of $C$, given by the assignment $a\mapsto C_{\leq a}$ (considered as a functor $A\to Cat_S$). ----------- Recall that the model structure on $Cat_S$ is [[combinatorial model category|combinatorial]] (and therefore [[cofibrantly generated]] and given as follows: Recall that there exists a small set of generators for Cof(S). Fix such a set of generators and call it $k$. Define $[1]_A\in Cat_S$ for any object $A\in S$ as the category with two objects $a,b$ such that $Map(a,a)=1_S$, $Map(b,b)=1_S$, $Map(a,b)=A$, and $Map(b,a)=\emptyset$. Define $[0]$ to be the $S$-category with one object $x$ such that $Map(x,x)=\emptyset$. Define the set $Q$ to be the set containing the unique map $\emptyset_{Cat_S}\to [0]$, and all of the induced maps $[1]_A\to [1]_B$ induced by generating cofibrations $(A\to B)\in k$ Then we define: Cof(Cat_S)=llp(rlp(Q)) We define the class of weak equivalences as follows: A weak equivalence of $S$-categories is an $S$-enriched functor $f:C\to D$ such that: - $Map_C(X,Y)\to Map_D(fX,fY)$ is a weak equivalence in $S$ - For every $Y\in D$, there exists an object $X\in C$ such that $Y$ is $S$-equivalent to $F(X)$ (homotopy essential surjectivity) It's well-known that these two classes define a model structure on $Cat_S$. (Recall that a morphism $g$ of an $S$-enriched category $C$ is called an equivalence if the corresponding map $hg$ in the Ho(S)-enriched category $hC$ is an isomorphism.) ------- Let $C$ be a model category. Recall that a natural transformation in a diagram category $C^D$ is called a weak equivalence (resp. projective fibration) if it is a weak-equivalence object-by-object (resp. a fibration object-by-object). The projective cofibrations are precisely those natural transformations with the left lifting property with respect to all natural transformations that are both fibrations and weak equivalences. If $C$ is left-proper combinatorial and $D$ is small, then $C^D$ with the projective model structure is likewise. -------- HTT Lemma A.3.5.9 states: For $A$ a poset and S an excellent model category, any $A$-filtered $S$-enriched category $C$ admits an $S$-enriched functor $f:C'\to C$ such that: - f is bijective on objects, and the morphisms $Map_{C'}(X,Y)\to Map_C(fX,fY)$ are trivial fibrations, - $C'$ is endowed with an induced $A$-filtration by composition $r\circ f$, - the induced diagram $a\mapsto C'_{\leq a}$ is a projectively cofibrant object of the model category $Cat_S^A$ This is given without proof, but there is a hint that one should use the small object argument. First of all, to show this, do we need to use the $S$-enriched version of the small object argument? If so, could someone give me a reference that states and proves this form of the [[small object argument]] (see the nLab page for an informal description, but no actual statement or proof)? If not, could someone break down exactly how to prove this?

   Before asking this question, we must recall some notions:

   ---

   Fix an excellent (therefore symmetric monoidal closed, combinatorial, and all monomorphisms are cofibrations (there’s more, but we don’t need it now)) model category $S$ (note: $S$ behaves in almost every way like the category of simplicial sets with the cartesian monoidal structure. If you’re not familiar with the definition of excellent, pretend that $S$ is the category of simplicial sets).

   We say that an $S$-enriched category $C$ is $A$-filtered, or equipped with an $A$-filtration, for a fixed poset $A$ if there exists a function $r:Ob(C)\to A$ such that if $r(X)\nleq r(Y)$, $Map_C(X,Y)=\emptyset$ where $\emptyset$ is the initial object of $S$.

   Any $A$-filtered $S$-category gives rise to a functor $A\to Cat_S$ as follows: Let $C_{\leq a}$ be the full subcategory of $C$ spanned by the set of objects $X$ such that $r(X)\leq a$. This gives a filtered diagram of subcategories of $C$, given by the assignment $a\mapsto C_{\leq a}$ (considered as a functor $A\to Cat_S$).

   ---

   Recall that the model structure on $Cat_S$ is combinatorial (and therefore cofibrantly generated and given as follows:

   Recall that there exists a small set of generators for Cof(S). Fix such a set of generators and call it $k$.

   Define $[1]_A\in Cat_S$ for any object $A\in S$ as the category with two objects $a,b$ such that $Map(a,a)=1_S$, $Map(b,b)=1_S$, $Map(a,b)=A$, and $Map(b,a)=\emptyset$. Define $[0]$ to be the $S$-category with one object $x$ such that $Map(x,x)=\emptyset$.

   Define the set $Q$ to be the set containing the unique map $\emptyset_{Cat_S}\to [0]$, and all of the induced maps $[1]_A\to [1]_B$ induced by generating cofibrations $(A\to B)\in k$

   Then we define:

   Cof(Cat_S)=llp(rlp(Q))

   We define the class of weak equivalences as follows:

   A weak equivalence of $S$-categories is an $S$-enriched functor $f:C\to D$ such that:

   • $Map_C(X,Y)\to Map_D(fX,fY)$ is a weak equivalence in $S$

   • For every $Y\in D$, there exists an object $X\in C$ such that $Y$ is $S$-equivalent to $F(X)$ (homotopy essential surjectivity)

   It’s well-known that these two classes define a model structure on $Cat_S$.

   (Recall that a morphism $g$ of an $S$-enriched category $C$ is called an equivalence if the corresponding map $hg$ in the Ho(S)-enriched category $hC$ is an isomorphism.)

   ---

   Let $C$ be a model category. Recall that a natural transformation in a diagram category $C^D$ is called a weak equivalence (resp. projective fibration) if it is a weak-equivalence object-by-object (resp. a fibration object-by-object). The projective cofibrations are precisely those natural transformations with the left lifting property with respect to all natural transformations that are both fibrations and weak equivalences. If $C$ is left-proper combinatorial and $D$ is small, then $C^D$ with the projective model structure is likewise.

   ---

   HTT Lemma A.3.5.9 states:

   For $A$ a poset and S an excellent model category, any $A$-filtered $S$-enriched category $C$ admits an $S$-enriched functor $f:C'\to C$ such that:

   • f is bijective on objects, and the morphisms $Map_{C'}(X,Y)\to Map_C(fX,fY)$ are trivial fibrations,
   • $C'$ is endowed with an induced $A$-filtration by composition $r\circ f$,
   • the induced diagram $a\mapsto C'_{\leq a}$ is a projectively cofibrant object of the model category $Cat_S^A$

   This is given without proof, but there is a hint that one should use the small object argument. First of all, to show this, do we need to use the $S$-enriched version of the small object argument? If so, could someone give me a reference that states and proves this form of the small object argument (see the nLab page for an informal description, but no actual statement or proof)?

   If not, could someone break down exactly how to prove this?

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 8th 2010
   • (edited Aug 8th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexNevermind, I got a response from Lurie by e-mail. I guess I'll expand it out and write this up on an nLab page, since I went to the trouble of writing all of the background up! =)

   Nevermind, I got a response from Lurie by e-mail. I guess I’ll expand it out and write this up on an nLab page, since I went to the trouble of writing all of the background up! =)