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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 11th 2010
   • (edited Dec 11th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexConsider the category $\Delta^+$ from the article on the [cartesian model structure](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#as_a_quasitopos_35). This category (I think!) satisfies some axioms of Cisinski that give the following result: The monomorphisms of $Psh(\Delta^+)$ are exactly $LLP(RLP(\mathcal{M}))$ where $\mathcal{M}$ is the small set $\{\partial \Delta^n\hookrightarrow \Delta^n\}\cup \{\Delta^1 \hookrightarrow \tilde{\Delta}^1\}$ where $\tilde{\Delta}^1$ is the presheaf representing the intermediate object $[\tilde{1}]$ (in the notation on the page, this is $[1]^+$. This is very cool if it's true. This comes from the section 8.1 of Cisinski's book. He proves a number of powerful theorems about "categories skelettiques", which are essentially a restricted class of generalized reedy categories where we can filter presheaves on them by certain canonical n-skeleton filtrations. In particular, the claim in the preceding paragraph depends integrally on $\Delta^+$ being normal and skelletique (this translates to "skeletal" in English, but it is not the same as the ordinary meaning of skeletal). Regarding the skelletique structure on $\Delta^+$, it is the same one as on $\Delta$, where the non-iso monomorphisms are the maps that raise degree, the non-iso surjections lower degree, and the isomorphisms do not change degree, and further, we define the skeletal degree of $[\tilde{1}]$ to be $2$. I have not yet checked that $\Delta^+$ is normal, but I suspect that it is. We denote by $(\mathfrak{G},\mathfrak{G})$ the contractible relative groupoid with two objects and equipped with its maximal relative structure. We have an evident "relative nerve" functor $\mathfrak{N}:RelCat\to Psh(\Delta)$, and we define $\mathcal{I}=\mathfrak{N}((\mathfrak{G},\mathfrak{G}))$. By the above result, $\mathcal{I}$ is injective with respect to the monomorphisms of $Psh(\Delta^+)$. The triple $(\mathcal{I}, \partial^0,\partial^1)$ where $\partial^i:\Delta^0\to \mathcal{I}$ are the evident inclusions. This is an [[interval object]], and the quadruple $(\mathcal{I}\times(-),\partial^0\times(-),\partial^1\times(-),\sigma\times(-))$ (where $\sigma$ is the terminal map) gives a functorial cylinder. We define $(\partial^0,\partial^1):\partial \mathcal{I} = \Delta^0\coprod \Delta^0\to \mathcal{I}$ to be the disjoint union of the evident vertices. Roughly following Lurie, let $S$ be the set consiting of the following four morphisms (we suppress the "flat" notation, since we find it cumbersome and confusing): $h^2_1:\Lambda^2_1\hookrightarrow \Delta^2$ $\{1\}:\Delta^0\hookrightarrow \tilde{\Delta}^1$ $q:(\Lambda^2_1)^#\coprod_{(\lambda^2_1)} \Delta^2\hookrightarrow (\Delta^2)^#$ $a:A\hookrightarrow (A,s_0A_0\cup \{f\})$ where $A$ is the quotient of $\Delta^3$ corepresenting the functor $$Hom(A,X)=\{x\in X_3, e\in X_1: d_1(x)=s_0(e) d_2(x)=s_1(e)\}$$ and $f$ is the image of $\Delta^{\{0,1\}}\hookrightarrow \Delta^3$ in $A$. (Note that the object here is still an object of $Psh(\Delta)$. The second coordinate is simply the value of the presheaf on $[\tilde{1}]$.) Note that we only take take $S^#$ for simplicial sets that are the nerve of a category. We are not using any of the structure on $Set^+_\Delta$. Now following Cisinski, we proceed to generate a generating set of anodyne morphisms for $T$ as follows: Define $\Lambda^0_\mathcal{I}(S,\mathcal{M})=S\cup\{\partial^i\wedge f : i\in\{0,1\}, f\in \mathcal{M}\}$ Where for $f:A\to B$ and $g:C\to D$, $f\wedge g=A\times D\coprod_{A\times C} B\times C\to B\times D$ is the lower righthand corner map. For any subset $U\subset Arr(\Delta^+)$, define $\Lambda^{+1}_\mathcal{I}(U)=\{(\partial^0,\partial^1)\wedge f: f\in U\}$ where $(\partial^0,\partial^1):\partial \mathcal{I}\to \mathcal{I}$. By recurrence, define $\Lambda^{n+1}_\mathcal{I}(S,\mathcal{M})=\Lambda^{+1}_\mathcal{I}(\Lambda^n_\mathcal{I}(S,\mathcal{M}))$, and finally, define $$\Lambda_\mathcal{I}(S,\mathcal{M})=\bigcup_{n\geq 0}(\Lambda^n_\mathcal{I}(S,\mathcal{M})).$$ Notice that this is a small set, and further, it is the smallest set stable under the operation $\Lambda^{+1}_\mathcal{I}$. By a theorem of Cisinski, $LLP(RLP(\Lambda_\mathcal{I}(S,\mathcal{M})))$ is the smallest class of anodyne morphisms (relative to the cylinder $\mathcal{I}$) containing the set $S$. It may be worth noting the definition of a class of anodyne morphisms $An_\mathcal{I}$ relative to cylinder $\mathcal{I}$: Anodyne morphisms: --- ------- $An_0:$ There exists a small set $S$ of monomorphisms such that $An_\mathcal{I}=LLP(RLP(S))$. $An_1:$ For any monomorphism $f:K\to L$, the smash products $\partial^i\wedge f \in An_\mathcal{I}$. $An_2:$ For any $f\in An_\mathcal{I}$, $(\partial^0,\partial^1)\wedge f\in An_\mathcal{I}$ (recall again that $(\partial^0,\partial^1):\partial \mathcal{I}\to \mathcal{I}$ is the canonical inclusion). By Cisinski's big theorem of chapter 1.3, these data generate a model structure.

   Consider the category $\Delta^+$ from the article on the cartesian model structure. This category (I think!) satisfies some axioms of Cisinski that give the following result: The monomorphisms of $Psh(\Delta^+)$ are exactly $LLP(RLP(\mathcal{M}))$ where $\mathcal{M}$ is the small set $\{\partial \Delta^n\hookrightarrow \Delta^n\}\cup \{\Delta^1 \hookrightarrow \tilde{\Delta}^1\}$ where $\tilde{\Delta}^1$ is the presheaf representing the intermediate object $[\tilde{1}]$ (in the notation on the page, this is $[1]^+$.

   This is very cool if it’s true. This comes from the section 8.1 of Cisinski’s book. He proves a number of powerful theorems about “categories skelettiques”, which are essentially a restricted class of generalized reedy categories where we can filter presheaves on them by certain canonical n-skeleton filtrations. In particular, the claim in the preceding paragraph depends integrally on $\Delta^+$ being normal and skelletique (this translates to “skeletal” in English, but it is not the same as the ordinary meaning of skeletal).

   Regarding the skelletique structure on $\Delta^+$, it is the same one as on $\Delta$, where the non-iso monomorphisms are the maps that raise degree, the non-iso surjections lower degree, and the isomorphisms do not change degree, and further, we define the skeletal degree of $[\tilde{1}]$ to be $2$. I have not yet checked that $\Delta^+$ is normal, but I suspect that it is.

   We denote by $(\mathfrak{G},\mathfrak{G})$ the contractible relative groupoid with two objects and equipped with its maximal relative structure. We have an evident “relative nerve” functor $\mathfrak{N}:RelCat\to Psh(\Delta)$, and we define $\mathcal{I}=\mathfrak{N}((\mathfrak{G},\mathfrak{G}))$. By the above result, $\mathcal{I}$ is injective with respect to the monomorphisms of $Psh(\Delta^+)$. The triple $(\mathcal{I}, \partial^0,\partial^1)$ where $\partial^i:\Delta^0\to \mathcal{I}$ are the evident inclusions. This is an interval object, and the quadruple $(\mathcal{I}\times(-),\partial^0\times(-),\partial^1\times(-),\sigma\times(-))$ (where $\sigma$ is the terminal map) gives a functorial cylinder. We define $(\partial^0,\partial^1):\partial \mathcal{I} = \Delta^0\coprod \Delta^0\to \mathcal{I}$ to be the disjoint union of the evident vertices.

   Roughly following Lurie, let $S$ be the set consiting of the following four morphisms (we suppress the “flat” notation, since we find it cumbersome and confusing):

   $h^2_1:\Lambda^2_1\hookrightarrow \Delta^2$
   $\{1\}:\Delta^0\hookrightarrow \tilde{\Delta}^1$
   $q:(\Lambda^2_1)^#\coprod_{(\lambda^2_1)} \Delta^2\hookrightarrow (\Delta^2)^#$
   $a:A\hookrightarrow (A,s_0A_0\cup \{f\})$

   where $A$ is the quotient of $\Delta^3$ corepresenting the functor

   $$Hom(A,X)=\{x\in X_3, e\in X_1: d_1(x)=s_0(e) d_2(x)=s_1(e)\}$$

   and $f$ is the image of $\Delta^{\{0,1\}}\hookrightarrow \Delta^3$ in $A$. (Note that the object here is still an object of $Psh(\Delta)$. The second coordinate is simply the value of the presheaf on $[\tilde{1}]$.)

   Note that we only take take $S^#$ for simplicial sets that are the nerve of a category. We are not using any of the structure on $Set^+_\Delta$.

   Now following Cisinski, we proceed to generate a generating set of anodyne morphisms for $T$ as follows:

   Define $\Lambda^0_\mathcal{I}(S,\mathcal{M})=S\cup\{\partial^i\wedge f : i\in\{0,1\}, f\in \mathcal{M}\}$

   Where for $f:A\to B$ and $g:C\to D$, $f\wedge g=A\times D\coprod_{A\times C} B\times C\to B\times D$ is the lower righthand corner map.

   For any subset $U\subset Arr(\Delta^+)$, define $\Lambda^{+1}_\mathcal{I}(U)=\{(\partial^0,\partial^1)\wedge f: f\in U\}$ where $(\partial^0,\partial^1):\partial \mathcal{I}\to \mathcal{I}$.

   By recurrence, define $\Lambda^{n+1}_\mathcal{I}(S,\mathcal{M})=\Lambda^{+1}_\mathcal{I}(\Lambda^n_\mathcal{I}(S,\mathcal{M}))$, and finally, define

   $$\Lambda_\mathcal{I}(S,\mathcal{M})=\bigcup_{n\geq 0}(\Lambda^n_\mathcal{I}(S,\mathcal{M})).$$

   Notice that this is a small set, and further, it is the smallest set stable under the operation $\Lambda^{+1}_\mathcal{I}$.

   By a theorem of Cisinski, $LLP(RLP(\Lambda_\mathcal{I}(S,\mathcal{M})))$ is the smallest class of anodyne morphisms (relative to the cylinder $\mathcal{I}$) containing the set $S$.

   It may be worth noting the definition of a class of anodyne morphisms $An_\mathcal{I}$ relative to cylinder $\mathcal{I}$:

   Anodyne morphisms:

   ---

   $An_0:$ There exists a small set $S$ of monomorphisms such that $An_\mathcal{I}=LLP(RLP(S))$.
   $An_1:$ For any monomorphism $f:K\to L$, the smash products $\partial^i\wedge f \in An_\mathcal{I}$.
   $An_2:$ For any $f\in An_\mathcal{I}$, $(\partial^0,\partial^1)\wedge f\in An_\mathcal{I}$ (recall again that $(\partial^0,\partial^1):\partial \mathcal{I}\to \mathcal{I}$ is the canonical inclusion).

   By Cisinski’s big theorem of chapter 1.3, these data generate a model structure.

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 11th 2010
   • (edited Dec 11th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexPossible pitfalls: ------ -------- The sets $\mathcal{M}$ and $S$ may not be correct. The set $\mathcal{M}$ may not be correct if the category $\Delta^+$ is not normal and skelletique. This is not too much of a problem, since the main issue here would be our choice of cylinder. Our choice of cylinder is particularly nice, but we could always fall back and use the [[Lawvere interval]] if this fails. It is a theorem that the class of monomorphisms of a category of presheaves on a small category admits a small set of generators (called a cellular model) $\mathcal{M}$ with accessible (compact, small, what have you) source and target (this is proven in Cisinski 1.2). More seriously, but still not too terrible, our set $S$ of Cisinski generators (these are generators of the set of generators of the set of anodyne morphisms!) may not contain enough morphisms. One solution would be to simply include all of Lurie's marked anodyne morphisms, but I am reasonably sure that the "anodyne closure" of $S$ will generate them, although I haven't checked. In particular, I'd be very happy if we could replace the last morphism, $a$, with something more intuitive. Interesting points ---- --- We have an evident realization functor $\Delta^+\hookrightarrow Set^+_\Delta$, which extends to a colimit preserving functor $Psh(\Delta^+)\hookrightarrow Set^+_\Delta$. I'm pretty sure that the realization functor preserves cofibrations (by exactness?), fibrant objects, and the cylinder object, but to get an idea of what else is true, I will probably have to follow Cisinski's construction of the model structure. One reason why this construction should be very powerful, if it works out, is that we will not only have lifted the model structure, but also the slicing structure (that is to say, this allows us to generate a model category over every object). Further, this construction should allow us to give a powerful generalization of Kan's $Ex^\infty$ based on the initial and terminal subdivision functors of Barwick and Kan (I hypothesize that if this whole crazy plan works, the correct subdivision functor for this anodyne structure should be either the terminal one or one of the conjugates of B-K's "relative double subdivision").

   Possible pitfalls:

   ---

   The sets $\mathcal{M}$ and $S$ may not be correct. The set $\mathcal{M}$ may not be correct if the category $\Delta^+$ is not normal and skelletique. This is not too much of a problem, since the main issue here would be our choice of cylinder. Our choice of cylinder is particularly nice, but we could always fall back and use the Lawvere interval if this fails. It is a theorem that the class of monomorphisms of a category of presheaves on a small category admits a small set of generators (called a cellular model) $\mathcal{M}$ with accessible (compact, small, what have you) source and target (this is proven in Cisinski 1.2).

   More seriously, but still not too terrible, our set $S$ of Cisinski generators (these are generators of the set of generators of the set of anodyne morphisms!) may not contain enough morphisms. One solution would be to simply include all of Lurie’s marked anodyne morphisms, but I am reasonably sure that the “anodyne closure” of $S$ will generate them, although I haven’t checked. In particular, I’d be very happy if we could replace the last morphism, $a$, with something more intuitive.

   Interesting points

   ---

   We have an evident realization functor $\Delta^+\hookrightarrow Set^+_\Delta$, which extends to a colimit preserving functor $Psh(\Delta^+)\hookrightarrow Set^+_\Delta$.

   I’m pretty sure that the realization functor preserves cofibrations (by exactness?), fibrant objects, and the cylinder object, but to get an idea of what else is true, I will probably have to follow Cisinski’s construction of the model structure.

   One reason why this construction should be very powerful, if it works out, is that we will not only have lifted the model structure, but also the slicing structure (that is to say, this allows us to generate a model category over every object). Further, this construction should allow us to give a powerful generalization of Kan’s $Ex^\infty$ based on the initial and terminal subdivision functors of Barwick and Kan (I hypothesize that if this whole crazy plan works, the correct subdivision functor for this anodyne structure should be either the terminal one or one of the conjugates of B-K’s “relative double subdivision”).

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 11th 2010
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexUh-oh. I think I might be screwed if the realization ends up being the same as the plus construction.

   Uh-oh. I think I might be screwed if the realization ends up being the same as the plus construction.