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.

  • $\begingroup$ Your proof that it works in Top with local homeomorphisms should extend to other infinitary superextensive sites without too much bother. For a finitary superextensive site it should work for finite J. I'll think about it a little more (but others may beat me to it) $\endgroup$ Nov 5, 2010 at 1:09
  • $\begingroup$ @David: Thanks- yes, I thought it might have to do with superextensiveness. Anyway, I never said I had a proof for $Top$, just that I know it's true. If you know a nice argument that will work for this and other superextensive sites, please enlighten me. I need to generalize this, but first, I need to understand the 1-categorical version a bit better. Thanks! $\endgroup$ Nov 5, 2010 at 1:20
  • $\begingroup$ I don't see why it is true for Top. It seems to me that you would need at least that the coprojections into the colimit of F form a covering family -- does that follow somehow from your assumption? $\endgroup$ Nov 5, 2010 at 4:27
  • $\begingroup$ @Mike: When I say that the diagram for $F$ consists only of elements of covering families, I mean also, each object $F(j)$ of $C/c$ is part of a covering on $c$. So, for $Top$, we have a diagram in local homeomorphisms over $X$ ($c$=$X$ now), i.e. a diagram in $Sh(X)$, hence its colimit in $Sh(X)$ exists, and since the etale space construction $Sh(X) \to Top/X$ is a left-adjoint, it preserves colimits- so the coprojections ARE local homeomorphisms. It's true in $Top$ because the (functor local homeormophisms) $\to$ $Sh(Top)/X$ is $j_!$ for $j$ the inclusion of opens of $X$ into $Top/X$ $\endgroup$ Nov 5, 2010 at 9:16
  • $\begingroup$ ...of course I mean, local homeomorphims over $X$ $\endgroup$ Nov 5, 2010 at 9:29

2 Answers 2


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.


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)

  • $\begingroup$ @Buschi: I'm having a little trouble understanding exactly what you are saying due to English, however, if I did indeed interpret what you meant correctly (linguistically), then I do not understand you mathematically. Where does the "natural" map $y(X) \to \varinjlim_I y(Xi)$ come from? I see a natural map in the other direction instead. I'm a little lost. $\endgroup$ Nov 7, 2010 at 20:27
  • $\begingroup$ @Carchedi. Of course you right, I edit and correct this mistake. $\endgroup$ Nov 8, 2010 at 19:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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