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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Created:
The dissolution locale of a locale is defined as the poset of its sublocales (equivalently: nuclei on ) equipped with the relation of reverse inclusion.
There is a canonical morphism of locales
such that the map sends an open to the open in given by the open sublocale of .
The map can be considered an analogue of the canonical map for a topological space , where is the underlying set of equipped with the discrete topology.
In particular, discontinuous maps could be defined as morphisms of locales , see Picado–Pultr, XIV.7.3.
Original reference:
Expository account:
created shifted tangent bundle because I thought somebody was asking about that on the blog, but now looking more closely I find that maybe nobody asked for that...
As an outcome of recent discussion at Math Overflow here, Mike Shulman suggested some nLab pages where comparisons of different definitions of compactness are rigorously established. I have created one such page: compactness and stable closure. (The importance and significance of the stable closure condition should be brought out better.)
I started a stub at affine logic as I saw the link requested in a couple of places.
The cut rule for linear logic used to be stated as
If and , then .
I don’t think this is general enough, so I corrected it to
If and , then .
the entry Galois theory used to be a stub with only some links. I have now added plenty of details.
I have tried to expand a bit the text at the beginning of the category:people entry Alexander Grothendieck, mention more of what his work was about, add more hyperlinks. It could still be much improved, but right now it reads as follows:
The french mathematician Alexandre Grothendieck, (in English usually Alexander Grothendieck), has created a work whose influence has shown him to be the greatest pure mathematician of the 20th century; and his ideas continue to be developed in this century.
Initially working on topological vector spaces and analysis, Grothendieck then made revolutionary advances in algebraic geometry by developing sheaf and topos theory and abelian sheaf cohomology and formulating algebraic geometry in these terms (locally ringed spaces, schemes). Later topos theory further developed independently and today serves as the foundation also for other kinds of geometry. Notably its homotopy theoretic refinement to higher topos theory serves as the foundation for modern derived algebraic geometry.
Grothendieck’s work is documented in texts known as EGA (with Dieudonné), an early account FGA, and the many volume account SGA of the seminars at l’IHÉS, Bures-sur-Yvette, where he was based at the time. (See the wikipedia article for some indication of the story from there until the early 1980s.)
By the way, in view of the recent objection to referring to people as “famous” in category:people entries: the lead-in sentence here is not due to me, it has been this way all along. One might feel that it should be rephrased, but I leave that to those who feel strongly about it.
I gave index an Idea-section.
In the course of this I created some stubby auxiliary entries, such as (in rapidly increasing order of stubbieness)
Added to Maslov index and to Lagrangian Grassmannian the following quick cohomological definition of the Maslov index:
The first ordinary cohomology of the stable Lagrangian Grassmannian with integer coefficients is isomorphic to the integers
The generator of this cohomology group is called the universal Maslov index
Given a Lagrangian submanifold of a symplectic manifold , its tangent bundle is classified by a function
The _Maslov index of is the universal Maslov index pulled back along this map
I added a Definition section to Burnside ring (and made Burnside rig redirect to it).
added statement of existence of linear extensions (here)
Will give this its own entry at linear extension of a partial order, for ease of referencing
[spam]
An early survey is
Added the Yoneda-embedding way to talk about group objects and hence supergroups.
added pointer to
which provides a wealth of computational details and illustrative graphics.
I wrote about Dmitri Pavlov’s concept of measurable locales.
I added some simpler motivation in terms of the basic example to the beginning of distributive law.