The sheafification of a presheaf on a site is often constructed in a twostep process $X^{++}$, where $X^+$ consists of matching families in $X$, is always separated, and is a sheaf if $X$ is separated. But the sheafification can also be constructed in a single step by looking at matching families over hypercovers. However, the only published reference I can find which mentions this latter fact is Higher Topos Theory (section 6.5.3). Is there a reference on "good old" 1sheaves which discusses sheafification via hypercovers?

3$\begingroup$ 1+ since I didn't know that this can be done with hypercoverings in one step. $\endgroup$– Martin BrandenburgMar 12, 2012 at 15:22

$\begingroup$ I'm no more too inside these questions, but may be: Lawrence Breen "On the Classification os 2gerbes and 2staks$ (Asterisque 225) p.38 p.38, 39 seems indicate how make a sheafification by hypercover language.... $\endgroup$– Buschi SergioApr 30, 2012 at 21:05

1$\begingroup$ I'm incredibly determined to give this post the answer it deserves... almost a decade later. $\endgroup$– cheyneMay 14, 2021 at 15:20
5 Answers
I don't know any reference where this is proven in elementary terms (although this can be done, of course). This is part of folklore since years (in spirit, this goes back to Verdier's formula in SGA 4 (exposé V) and in Ken Brown's thesis), but the only explicit reference I know is Proposition 7.9 (for $n=0$) in the paper
Dugger, Hollander and Isaksen, Hypercovers and simplicial presheaves, Math. Proc. Camb. Phil. Soc. 136 (2004), 951. (See here for a preprint version.)

1$\begingroup$ Thanks. At least this is a proof, but I was hoping for something written elementarily for 1sheaves. $\endgroup$ Mar 12, 2012 at 20:49
Sergio just brought into my attention this question. The definition of locally compatible family says exactly that the family is compatible over a hypercover refinement. So the one step construction in Yuhjtman thesis is just the onestep hypercover construction.
However the hypercover in question is simply determined by a cover of the 1simplices
$U_i \times_U U_j$ of the cover $U_i \to U$, so it seems unnecessary to mention the hypercover notion. I discover this onestep construction a long time ago, and at that time I was ignorant of the hypercover notion, which as we know, is much more complicated than just the particular case determined by a cover of the 1simplices.
Eduardo J. Dubuc


$\begingroup$ Oh, and I took the liberty of editing your $x$ to $\times$ for readability. $\endgroup$– David Roberts ♦May 1, 2012 at 1:01

1$\begingroup$ Thanks! Of course for dealing with ordinary sheaves, it is not strictly necessary to speak about hypercovers, but when thinking (as I like to do) about higher sheaves as well, I find it helpful to talk about 1sheaves in language which generalizes easily. $\endgroup$ May 1, 2012 at 4:00
One way of constructing the associated sheaf in one step is written here: http://cms.dm.uba.ar/academico/carreras/licenciatura/tesis/yuhjtman.pdf (in spanish) page 19, (3.2). The key idea (due to Eduardo Dubuc) is to consider "locally compatible families" instead of just "compatible families".

$\begingroup$ hi, I humbly think it would be a good idea if you could expand a little your answer to give a sketch of the argument (after all it is your thesis! :) It might also be useful as to highlight the similarities/differences with the hypercovers approach. $\endgroup$ Apr 30, 2012 at 20:54

$\begingroup$ This proof has no surprising ideas other than the definition of "locally compatible families". From there it is like the usual ++ construction. I'm not familiar with the hypercovers approach. $\endgroup$ Apr 30, 2012 at 21:10
let $F$ a presheaf on a topological space $X$ (all is straight generalizable to a general site).
THen the separate presheaf $L(F)$ associate is definited as follow:
$L(F)(U)$ is the colimit of the sets $C^>(R, F)$ where: $C$ is the category of the opens of $X$, $C^>$ category of presheaves on $C$, and $R\subset h_U$ is a $X$cover (covering cribles) of $U$, and the colimts is about all $X$cover of $U$ and its inclusion morphisms.
Then (by Yoneda lemma, and natural representation os a presheaf as the colimits of representable by the comma category on it ) we can represent the elements of $L(F)(U)$ as a class of equivalence of families $[(U_i, x_i)_{i\in I}]$ with $U_i$ form a open covering of $U$, $x:i\in U_i$ and identify two of these family: $(U_i, x_i)_{i\in I}$ and $(V_j, y_j)_{j\in J}$ if $\forall i, j\in I\times J: {x_i}_{U_i\cap V_j}= {y_j}_{U_i\cap V_j}$.
as in the usual theorem follow that $L(F)$ is a separate presheaf, is a sheaf if $F$ is separate, is isomorphic to $F$ is $F$ is a sheaf, and $LL(F)$ is the sheaf associate with the canonical universal property.
Give a example (ad hoc) of a preshaf $F$ such that $L(F)$ isnt a sheaf (neccessarly $F$ isnt separate).
Let the topolgy $\tau_X=${$X, U, V, A, B, A\cap B, \emptyset$} with $X=U\cup V$ and $U\cap V=A\cup B$.
Let $F(X)=\emptyset=F(\emptyset)$, $F(U)=${$a$}, $F(V)=${$b$} with $a_{U\cap V}\neq b_{U\cap V}$ but $a_A=b_A,\ a_B=b_B, $ then consider $\alpha:=[(U, a)]\in LF(U),\ \beta:=[(V, b)]\in LF(V)$ we have that
$\alpha_{U\cap V}= \beta_{U\cap V}=[${$(A, a_A), (B, b_B)$}$]$
but $\alpha$ and $\beta$ cannot come from a (global) element of $F(X)$.
THis example (I hope ) explain the the difficulty that prevents $ L (F) $ to be a sheaf.
for if $X=U\cup V$, in general gived $s\in L(F)(U)$, $s=[(U_i, x_i)_I]$, and $t\in L(F)(V)$, $t=[(V_j, y_j)_J]$,
with $s_{U\cap V}= t_{U\cap V}$ this last condiction could use a refiniment of $(U_i\cap V)_I$ and $(V_j\cap U)_J$ and could be that ${s_i}_{U_i\cap V_j} \neq {t_j}_{U_i\cap V_j}$ but these are equal on a more strink refiniment.
Now is instead of the coverings, we use (3level I think) hypercovering (see for example Definition 2.4 on Lawrence Breen "On the Classification os 2gerbes and 2staks" (Asterisque 225) p.38, 39) we have a more rich representation: $[(U_i, x_i), (U_{i,j, a})_{a\in Ai,j}]$ where for $i, j\in I$ we have
$U_i\cap U_j=\bigcup_{a\in Aij}U_{i,j,a}$ and ${x_i}_{Uija}={x_j}_{Uija}\ a\in Aij$.
with the natural equavalence relation "..agree on a common refiniment".
In this way the above difficulties are overcome, and has a sheaf directly to the first step.
You may get lucky with Kashiaware and Schapira's newer book
Categories and sheaves. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 332. SpringerVerlag, Berlin, 2006.

1$\begingroup$ Can you give a precise reference in this book? Or is it just a guess? $\endgroup$ Mar 12, 2012 at 19:21

$\begingroup$ I looked there, but I couldn't find it. They discuss sheafification mostly in terms of local isomorphisms, which I suppose might be unravellable to say something about hypercovers, but it doesn't seem like it would be much less work than proving it directly. $\endgroup$ Mar 12, 2012 at 20:47