Question

The sheafification of a presheaf on a site is often constructed in a two-step 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" 1-sheaves which discusses sheafification via hypercovers?

Answer 1

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), 9-51. (See here for a preprint version.)

Answer 2

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".

Answer 3

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 one-step hypercover construction.

However the hypercover in question is simply determined by a cover of the 1-simplices $U_i \times_U U_j$ of the cover $U_i \to U$, so it seems unnecessary to mention the hypercover notion. I discover this one-step 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 1-simplices.
Eduardo J. Dubuc

Answer 4

You may get lucky with Kashiaware and Schapira's newer book

Categories and sheaves. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 332. Springer-Verlag, Berlin, 2006.

Answer 5

let $F$ a presheaf on a topological space $X$ (all is straight generalizable to a general site).

THen the separate pre-sheaf $L(F)$ 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 (3-level I think) hypercovering (see for example Definition 2.4 on Lawrence Breen "On the Classification os 2-gerbes and 2-staks" (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.