Talk:Grothendieck topology

message: I'm not an expert, but I'd suggest to replace the term 'open immersion' with 'inclusion', since the former one is used rather in the context of manifolds.
:Please sign your posts, Mr. 128.100.216.221.  Use four tildes (the ~ thingy).
:I wrote "open immersion" because I come out of the world of algebraic geometry.  An open immersion in algebraic geometry is the inclusion map from an open subset U of an algebraic variety to the entire variety, and it's the correct term in that context; but I didn't notice that the term was confusing in the context of manifolds!  I'm not convinced, however, that it would be appropriate to change "open immersion" to "inclusion" everywhere.  I'm not sure what use Grothendieck topologies have for people interested purely in manifolds; so far as I'm aware, in practice they only turn up via schemes or rigid analytic spaces.  Manifolds already come with good (topological space-style) topologies, and so do spaces of functions like C^\infty(M).  Only crazy people like me talk about Grothendieck topologies, and for us, "open immersion" is the correct term.
:For the moment, I've added a short clarification to the article.  If that's not sufficient, please edit or make a suggestion.  141.211.62.20 01:55, 29 January 2007 (UTC)

message: I have reverted the sentence "Grothendieck topologies are not equivalent to the classical notion of topological spaces." to "Grothendieck topologies are not comparable to the classical notion of topological spaces."  My reasoning is as follows: Both sentences state that the two concepts are inequivalent, but only the latter indicates that neither concept is a generalization of the other.  Old versions of the article hinted that Grothendieck topologies were more general, but this is false for pathological topological spaces, and I don't want anyone to come away from this article with that misconception.  141.211.62.20 00:49, 13 February 2007 (UTC)
replies: text: I don't like "comparable" because, under its plain meaning, the theory of Grothendieck topologies exists to be compared to classical topology.  What do you think of my rewrite?  Changbao 06:44, 13 February 2007 (UTC)
replies: text:  Very good, better than before.  I copyedited it a little—I thought it was important to mention that topological spaces give sites, not just Grothendieck topologies, since I thought that might confuse a beginner.  Also, I wouldn't call the differences between the two theories minor.  Yes, for sober topological spaces they're the same, but the setup for the theories is completely different, and there are lots of useful Grothendieck topologies which don't come from topological spaces, such as the étale and flat sites.  Feel free to refine the paragraph even more.  141.211.62.20 00:32, 14 February 2007 (UTC)

text: BTW why the indiscrete topology does not come from a site?  —Preceding unsigned comment added by 128.100.216.180 (talk) 23:31, 5 March 2009 (UTC)
replies: text: It's not a sober space (if it has more than one point).  For, suppose that x and y are points of the space.  Then the closure of either x or y is the intersection of all closed set containing them; and the only non-empty closed set is the whole space.  So both x and y are generic points of the space. Ozob (talk) 16:24, 6 March 2009 (UTC)
thread_title:  "Comparable" vs. "equivalent" 

message: Some (other) anonymous user tried to define a morphism of sites.  The definition used, though obvious, is incorrect, so I've removed it.  The correct notions are continuity  and cocontinuity .   gives an example of a functor between two sites which preserves covering but which is not continuous, i.e., does not preserve sheaves, and that's the condition which is really useful.  141.211.120.63 19:11, 8 June 2007 (UTC)
thread_title:  Morphisms of sites 

message: Why does this page only define Grothendieck pretopologies for categories with pullbacks? Couldn't we get rid of (PT 0) and replace (PT 1) with 
(PT 1') If  \{ f_i: X_i \rightarrow X | i \in I\}  is a covering family of X, then for any morphism  g: Y \rightarrow X there is a covering family  \{ h_j: Y_j \rightarrow Y | j \in J  \} such that for each j, g \circ h_j factors through some f_i.
Then in the case that C has pullbacks, we can derive (PT 1). When we don't have pullbacks we still get a Grothendieck topology by defining covering sieves to be those which contain some covering family.
Steven Gubkin (talk) 21:26, 14 November 2008 (UTC)
replies: text: Yes, you could do that, but it's not standard. (Remember, we don't make innovations here on WP, we just quote what other people have done.) I think also you run into the problem that there's no advantage to doing things the way you're proposing; rather than use (PT 1'), one might as well work with sieves and topologies.  Most of the time, people use pretopologies because they find them a little easier, and with (PT 1') you lose some of that. Ozob (talk) 00:42, 15 November 2008 (UTC)
thread_title:  Definition of a grothendieck pretopology 

message: Hello there,
I found this article as an excellent explanation of the Grothendieck topology subject. I would like to suggest you to add the book of S. MacLane and M. Moerdijk (Sehaves in geometry and logic, Springer) as a reference, I found it quite useful and maybe it would help others too!
Keep up the good job fella!
--Dieu reconnaîtra les siens 10:58, 12 February 2009 (UTC)
thread_title:  Comment 

message: I think the explanation of (T 1) goes the wrong way: The maximal sieve is the one generated by the minimal cover, so (T 1) corresponds rather to the idea that each set covers itself.  — Preceding unsigned comment added by 82.20.208.189 (talk) 17:48, 11 February 2018 (UTC)
thread_title:  (T 1) explanation 

message: The "integral" word links to "integer". Shouldn't it be "integer coefficients" rather than "integral coefficients"? TricksterWolf (talk) 12:41, 12 February 2021 (UTC)
thread_title:  Integral?