Question

Hi there, I'm trying to sheafify a constant presheaf on a site, I went to http://ncatlab.org/nlab/show/sheafification, but can't understand the notation in the equation for W (in the proof for existence)

W = {S(U_i) := lim... }.

Can anyone help me figure out the notation? With respect to what is the limit taken? I'm still trying to figure out the whole sheafification process but so far I have like 2 or 3 books on the subject and none of them has been exactly helpful or straightforward for my liking, I need this result for a proof I'm working on and am not exactly an expert in Topos theory, I just need this result, hope you guys can help.

Answer 1

The $\lim_\to$ in that formula denotes the colimit over that diagram of two parallel morphisms right after it. So it's the coequalizer of these two morphisms. This is just a very explicit way (or maybe a very implicit way? :-) to write out the sieve that corresponds to a given cover in the site (it's the presheaf that sends any test object to the set of ways of homming that test object into the object being covered that factor through the cover)

As the text around the formula meant to indicate (I have now tried to clarify this further in the entry) the details for this are discussed (in quite some pedestrian detail, I think) at the entry on sieves

http://ncatlab.org/nlab/show/sieve

and that on sheaves

http://ncatlab.org/nlab/show/sheaf

Probably such questions on nLab entries are best posted (and most welcome) over at the nForum

http://www.math.ntnu.no/~stacey/Mathforge/nForum/