Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $A$ be a small category, and let $X:=Psh(A)$ denote the category of presheaves on $A$. It is a theorem that for any such category $X$, there exists a small set $M$ of monomorphisms admitting the small object argument such that $LLP(RLP(M))$ is exactly the class of all monomorphisms of $X$. Recall that a separated segment (a separated interval) is a triple $(I,\partial^0,\partial^1)$ where $\partial^i:*\to I$ (where $*$ denotes the terminal object) and such that the pullback of the diagram $\partial^0:*\hookrightarrow I\hookleftarrow *:\partial^1$ is the empty presheaf.

This triple defines a functorial cylinder $(I\times(-),\partial^0\times id_{(-)}, \partial^1\times id_{(-)},\sigma\times id_{(-)})$ where $\sigma:I\to *$ is the terminal map. By abuse of notation, we will write for any object $P$ in $X$, $\partial^0_P:=\partial^0\times id_P, \partial^1_P:=\partial^1\times id_P$ and $\sigma_P:= \sigma\times id_P$. Since $X$ is a presheaf topos, we can see easily that given any monomorphism $K\to L$ in $X$, the square:

$$\begin{matrix}K&\hookrightarrow &L\\ \downarrow&&\downarrow\\ I\times K&\hookrightarrow&I\times L\end{matrix}$$

(where the vertical arrows are the induced maps $\partial^0_K$ and $\partial^0_L$, or $\partial^1_K$ and $\partial^1_L$) is cartesian and is composed exclusively of monomorphisms. Because of this very fine property, we may define $I\times K \cup \{i\}\times L$ to be the subobject of $I\times L$ given by the apparent inclusion of the pushout where $i$ depends on the $\partial^i$ appearing in the above diagram. Adding to our list of suggestive notation, we define the map $\partial I:= *\coprod *$ considered as a subobject of $I$ by the canonical map from the coproduct $(\partial^0,\partial^1)$ (similarly, we define $\{0\}$ and $\{1\}$ to be the subobjects corresponding to the obvious maps (in this notation, $\partial I = \{0\}\coprod \{1\}$). We denote the previously mentioned inclusion $(\partial^0,\partial^1)$ by $b:\partial I\hookrightarrow I$, and as with the other distinguished maps, putting a subscript gives the obvious piece of the natural transformation.

Given any two morphisms $f:A\to A',g:B\to B'$ in $X$, define their smash product $f\wedge g:A\times B'\coprod_{A\times B} A'\times B \to A'\times B'$. Note that the smash product gives a monoidal product on $Arr(X)$ (the unit being the inclusion of the empty presheaf into the terminal one).

Given a separated segment $I$ in $X$, define a class of anodyne morphisms relative to $I$ to be a class $An$ of monomorphisms of $X$ satisfying the following three conditions:

$An_0:$ There exists a small set $S$ of monomorphisms such that $An=LLP(RLP(S))$.
$An_1:$ For any monomorphism $f:K\hookrightarrow L$, the smash products $\partial^i\wedge f$ are elements of $An$ for $i=0,1$.
$An_2:$ For any $f\in An$, the smash product $b\wedge f$ is an element of $An$ (recall again that $b:\partial I\to I$ is the canonical inclusion).

Question: Given any category of presheaves $X$, any separated segment $I$ on $X$, and any class of morphisms $An$ anodyne with respect to $I$, is it the case that given any monomorphism $f$ in $X$ and any anodyne morphism $g$ in $An$ that $f\wedge g\in An$? If this is true would you mind sketching a proof?

share|improve this question
    
In your second sentence you look like you need $M = S$ :) –  David Roberts Dec 13 '10 at 6:08
    
I picked a different variable and forgot to change all instances. Thanks! –  Harry Gindi Dec 13 '10 at 6:16
4  
I think that explaining where this comes from (namely, Cisinski's work, right?) and giving a reference could help the non-expert but interested readers and thus improve the question (just my two cents). –  Jonathan Chiche Dec 13 '10 at 6:25
1  
"I have my screen oriented vertically" Ok now I get it. –  Martin Brandenburg Dec 13 '10 at 9:11
1  
I remember Maltsiniotis is thanked in the "Préambule" of Cisinski's book: « en particulier, il a amélioré la définition d'extension anodine en dégageant une axiomatique simple ». –  Jonathan Chiche Dec 13 '10 at 9:19

1 Answer 1

up vote 0 down vote accepted

This condition is equivalent to the following condition:

The class $\mathrm{An}=\mathrm{An}(\Lambda_I(S))$ of anodynes generated by a small set $S$ of monomorphisms with respect to the cylinder $I$ is closed under finite Cartesian products with objects of $X=\mathrm{Psh}(A)$. That is, for any anodyne $f:K\hookrightarrow L$, and any object $B$ in $X$, the map $B\times f: B\times K\hookrightarrow B\times L$ is also anodyne, or equivalently, that the $A$-localizer $W(An)=W_I(S)=W(\Lambda_I(S))$ (which is accessible since $S$ is a small set) generated by $An$ is a Cartesian localizer.

This follows from Corollary 1.3.58, which in particular, says that $An$ has the following property:

If $u,v$ are composable monomorphisms in $X$ with composite $vu$ such that $u\in An$ and $vu\in An$, then $v\in An$.

Suppose that $f:K\hookrightarrow L$ is anodyne and $g:B\hookrightarrow C$ is any monomorphism.

We have the evident diagram:

$$\begin{matrix}B\times K&\hookrightarrow &C\times K\\ \downarrow&&\downarrow\\ B\times L&\hookrightarrow&C\times L\end{matrix}$$

In which the vertical maps are anodyne (by assumption).

Pushing out with respect to the top left corner, we get another diagram:

$$\begin{matrix}B\times K&\hookrightarrow &C\times K\\ \downarrow&&\downarrow\\ B\times L&\hookrightarrow&P\end{matrix}$$

and a unique map $P\hookrightarrow C\times L$ making things commute as they should. Notice now that $$C\times K \hookrightarrow C\times L = C\times K \hookrightarrow P \hookrightarrow C\times L.$$ However, the LHS is anodyne, and the map $B\times K\hookrightarrow P$ is anodyne as well, being a pushout of an anodyne map. Restating this, we see that $vu$ is anodyne and that $u$ is anodyne. Therefore, it follows that $v=f\wedge g:P\hookrightarrow B\times L$ is itself anodyne.

Also, it turns out that this answer holds in the slightly more general case whenever $I$ is a "good functorial cylinder" for $X$.

As the localizer $W(An)$ is accessible (by the axioms for a class of anodynes) and therefore is the class of weak equivalences for a model structure on $X$ where the cofibrations are the monomorphisms, this is equivalent to asking that the associated model structure is Cartesian-closed, since it implies that the fibrant objects are precisely those objects $B$ such that for every anodyne morphism $K\hookrightarrow L$, the induced map on internal function objects $B^L\to B^K$ is a trivial fibration in the associated model structure.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.