Talk:Gluing axiom

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I'm missing the definition of a fine sheaf here (and/or on sheaf (mathematics)). I think I could manage to write it, but as I'm not a guru, I prefer leaving this to s.o. "serious"... — MFH: Talk 20:44, 13 May 2005 (UTC)

[edit] wording

I think this page strongly needs several improvements.

First, the "zeroeth section" isn't a "gentle introduction" at all. This page severely lacks a somehow understandable introduction for people less familiar with formalism. Usually I advocate for rigourous definitions instead of too much handwaving (if I think of polynomial or so), but here I think it's too abstract right from the beginning. "Gluing axiom" has a quite easily understandable meaning in usual categories, which could be given in the introduction, e.g. "... the G.A. is the abstract concept generalizing the property of... (functions can be extended to a global section)..."


Concerning the current wording:

... the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy...

why "is introduced to define", and not simply "states" or so?


Here Open(X) is the partial order of open sets of X ordered by inclusion maps;

I think it should read "the partially ordered set of...": the 'partial order itself is the relation, thus a subset of the cartesian product.


As phrased on the sheaf page, there is a certain axiom that F must satisfy...

this is not clear: what "certain axiom", and: "F must satisfy", in order to be what ?


   F(X) is the subset of F(U)×F(V) with equal image in F(W).

"equal image": equal to what? image under what map?

Of course this is understandable if one already knows what is explained, but the page should be helpful for someone who does not know what it is. — MFH:Talk 13:22, 21 March 2006 (UTC)

[edit] colimits turn into limits?

It is not true that the sheaf functor turns colimits of all small diagrams D into limits of the corresponding diagrams. I fixed this in the article.

To see why this is so, take an open set W which is contained in the intersection of some two open sets U and V. the colimit of the diagram is U \cup V, regardless of what W is. We can even take W to be the empty set. So assume W is the empty set. The image of the diagram under F has F(U) and F(V) with arrows to the F(W), which is a set containing only the empty function. the limit of this diagram is the product of F(U) and F(V), which is in general different from F(U \cup V) - the set of pairs of sections which agree on U \cap V.

Essentially we "threw out" the compatibility requirement by shrinking W...

Amitushtush (talk) 16:42, 9 April 2008 (UTC)