Colimits of covers - MathOverflow

Colimits of covers

2010-11-04T23:25:23Z

Suppose I have category $C$ equipped with a Grothendiek pretopology of covers, and let $y:C \to Sh(C)$ be the Yoneda embedding into sheaves and $y/c:C/c \to Sh(C)/y(c)\cong Sh(C/c)$. How can I show that if $F:J \to C/c$ is any functor such its diagram consists only of elements of covering families, then:

$\left(y/c\right) \circ \varinjlim F = \varinjlim \left(y/c\right) \circ F$?

For example, this is true if $C = Top$ (topological spaces) and we equip it with the Grothendieck pretopology of jointly surjective local homeomorphisms, but I believe it should hold in greater generality.

Answer by David Carchedi for Colimits of covers

2010-11-05T17:29:18Z

This answer is not meant to discourage others from giving a complete answer, but only to help get towards a full one:

(Thanks to Urs Schreiber for helping me work this out)

For $C=Top$, you can prove this as follows. To show that $Et \to Sh(Top)$ (where $Et$ is topological spaces with only local homeomorphisms) preserves all colimits, it suffices to show that it preserves all coproducts, and also all coequalizers. Coproducts is easy- any cover of a disjoint union of spaces is the same as cover of each of them separately. Now suppose that $A \rightrightarrows B \to C$ is a coequalizing diagram in $Et$. Then, $B \to C$ is surjective and a local homeomorphism, hence a cover in $Top$. Let $C'$ denote the coqualizer of this diagram after being embedded into sheaves. There is an canonical map $C' \to y(C)$ induced from the image of the cocone on $C$ under $y$. I will show there is a map in the other direction, which I claim is an inverse for it:

Let $p_A$ and $p_B$ be the components of the cocone over $C'$. Consider the cover $B \to C$. I claim that $p_B$ is descent data for $C'$ for this cover.

To see this, note that there is a canonical map $e:A \to B \times_{C} B$ which is surjective and a local homeomorphism (by 2/3 and the fact that local homeomorphisms are stable under pullbacks). This implies that after composition with Yoneda, it becomes an epimorphsm. However the two maps $p_B \circ pr_1$ and $p_B \circ pr_2$ clearly agree after precomposing with $e$- but $e$ is epi, therefore they agree already- so $p_B$ is descent data.

So we get a map $C \to C'$, which I claim is inverse to the former map $C' \to C$. It's pretty easy to see how to adapt this to the "sliced" version as well.

Answer by Buschi Sergio for Colimits of covers

2010-11-07T18:06:40Z

I state what above in your intervention.

And I state what follow:

1] Let $\tau$ the Grothendieck topology on $\mathscr{C}$. Gived a sieve $R\subset X$ (considering it as a full subcategory of $\mathscr{C}\downarrow X$ or a subobjet of $h_X$) we call it a $\tau$-covering (of $X$) if for any sheaf $S$ the restriction morphism $R^\star: S(X)\cong Shv(X, S)\to Shv(R, S) \cong >{\underrightarrow{lim}}_{(y\to X)\in R} S(Y) $ is a isomorphism ($Shv$ mean “sheaves”).

This is equivalent to say one of the following two equivalent condiction:

a) $\iota: R \subset X$ is a Isomorphism in the category $Shv(\mathscr{C}, \tau)$

b) The image of $R \subset_{full} \mathscr{C} \downarrow X $ in $Shv(\mathscr{C} , \tau)$ describes a colimit cocone of $X\in Shv(\mathscr{C} , \tau)$.

The class of all $\tau$-covering define e Grothendieck topology $\widetilde{\tau} $ such taht $Shv(\mathscr{C} , \widetilde{\tau})=Shv(\mathscr{C} , \tau)$, and is the bigger topology with this propriety.

b’) Gived a family $\mathcal{F} =(f_i: X_i\to X)$. The sieve generated is $\tau$-covering iff : completing $\mathcal{F}$ by all couple af pullback $X_i\times_X X_j\ i,j\in I$ and let $\mathcal{F’}$ the enriched family (observe that the first inclusion $\mathcal{F'}\subset R \subset \mathscr{C} \downarrow X$ is final) then the image of $\mathcal{F’}$ in $Shv(\mathscr{C} , \tau)$ is a colimit cocone (in literature find also a "Pullback invariant condition", in this case this follow automatically) .

Now your request is the following condition:

give a diagram $(X_i \xrightarrow{x_i} C)_{i\in I}$

(dont write transitions morhisms) and let $X:= {\underrightarrow{lim}}_{I} X_i$ in $\mathscr{C}$, the natural morphism $y(X) \to {\underrightarrow{lim}}_{I} y(X_i) $ is a isomorphism in $Shv(\mathscr{C} , \tau)$.

Infact we state that:

2] Considering that in any category $\mathscr{C}$ the proiection funtor $\pi : \mathscr{C}\downarrow X \to \mathscr{C} $ create colimits (i.e. make o colimit in the comme $\mathscr{C}\downarrow X$ is “the some” that make the some colimit in $\mathscr{C}$).

Then if in your data the object $C$ isnt fixed but generic your request is equivalent to the follow:

give a diagram $(X_i)_i$

and let $X:= {\underrightarrow{lim}}_{I} X_i$ the natural morphism $y(X) \to {\underrightarrow{lim}}_{I} y(X_i) $ is a isomorphism in $Shv(\mathscr{C} , \tau)$. (you can put $C:= X$).

Then form 1-(b’) above this is neccessary that the colimit cocone $X_i \to X$ generate a $\tau$-covering, then we can state the condiction as follow:

give a colimit cocone $(X_i \to X)_{i\in I}$

and suppose that it generate a $\tau$-sieve, then ${\underrightarrow{lim}}_{I} y(X_i) \to y(X)$ is a isomorphism in $Shv(\mathscr{C} , \tau)$?

But this is equivalent to the condiction $\tau$ is sub-canonical, i.e. any representable presheaf $h_X$ is a sheaf or equivalently any cover is a colimit (or precover (completated by pullbak’s) if we start from a pretopology).

Of course this happen for topological covering (any open topological covering is a colimit too)