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I started an entry for Rijke’s join of maps. How do you place the lower?
I have added both to proof and to experiment pointer to
with the quote (from p. 2):
we claim that the role of rigorous proof in mathematics is functionally analogous to the role of experiment in the natural sciences
added statement and proof that sequentially compact metric spaces are equivalently compact metric spaces
added statement and proof at continuous images of compact spaces are compact. Cross-linked with extreme value theorem and other related entries.
added a proof at compact subspaces of Hausdorff spaces are closed
added statement-entry closed subspaces of compact Hausdorff spaces are equivalently compact subspaces.
with proof by pointing to the two directions:
Cross-linked with compact space, closed subspace and also with Heine-Borel theorem
created the line with two origins
I added a remark about the contravariant Yoneda embedding to the page on the Yoneda embedding.
It’s pretty elementary, but I think worth mentioning for those new to category theory that this is just the Yoneda embedding of the opposite category .
added statement and proof that sequentially compact metric spaces are totally bounded
added statement and proof to Lebesgue number lemma.
did the same for sequentially compact metric spaces are equivalently compact metric spaces but now the Lab seems to have gone down before I could submit the edit.
I move null set to null subset and added more about how these are defined in unusual contexts.
I added the theorem that complete norms on a real vector space are unique (up to topological equivalence) at norm#dreamUnique. (This is false in classical mathematics, of course, but it’s true in dream mathematics.) Also true for F-norms.
expanded endomorphism operad
(it’s still a bit rough, but I am a bit rushed and have no time to polish)
I added to uniformly regular space a definition of “uniform apartness space” and a proof that under uniform regularity, these coincide with ordinary uniform spaces. I think this is interesting because it seems to be one of the purposes of uniform regularity (and local decomposability).
I rescued an empty page and wrote F-norm.
This theorem, with a constructive proof, is now at convergence space. (The usual proof in undergraduate metric-space theory uses both excluded middle and countable choice1, so I wrote this mostly to verify that it is actually perfectly constructive in the general setting.)
ETA: And the straightforward generalization to nonsequential spaces would use choice of arbitrarily high cardinality. ↩
Back in 2015, Bas Spitters wrote filter space. These are even more general than convergence spaces! (In a filter space, even if two filters both converge to the same point, their intersection might not.) I've put in the definition from the cited paper by Martin Hyland.
I added a note about this to Grothendieck topos. Also I fixed a mistake that Mike found in the classification at pretopos.
Since I wondered what they were, I started an entry Gorenstein ring spectrum, which then needed Gorenstein ring. Not sure I’m much the wiser as to their importance. There should be a lot to say about related duality.
created exact (infinity,1)-functor
created category of operators
I realized that we had a stub entry “configuration space” with the physics concept, and a stub entry “Fadell’s configuration space” with the maths concept, and no interrelation between them, also without any examples. So I created a disambiguation page
and then
but I also left
separate for the moment, thinking that in principle the term in matematics may be understood more generally, too. But maybe something should be merged here.
I added the example of the unordered configuration space of as a model for the classifying space for the symmetric group to the relevant entries. But otherwise they do remain stubby, alas.
created law of double negation with just the absolute minimum. Added a link from double negation, but nothing more.
In the nLab article on the universal enveloping algebra, the section describing the Hopf algebra structure originally stated that “the coproduct is induced by the diagonal map .”
I assume that this is a mistake, and I have since changed the coproduct to a product . However, I don’t know a great deal about Hopf algebras, so please correct me if I’ve made a mistake here.
In line with the “pages named after theorems” philosophy, I’ve created toposes are extensive, including in particular the (somewhat hard to track down) constructive proof that a cocomplete elementary topos is infinitary extensive.
Urs started pointless topology, and I continued it.
I have added to Postnikov tower paragraphs on the relative version, (definition and construction in simplicial sets).
I also added the remark that the relative Postnikov tower is the tower given by the (n-connected, n-truncated) factorization system as varies, hence is the tower of n-images of a map in . And linked back from these entries.
I finally gave spectral super-scheme an entry, briefly stating the idea.
This goes back to the observation highlighted in Rezk 09, section 2. There is some further support for the idea that a good definition of supergeometry in the spectrally derived/ context is nothing but -geometry over even periodic ring spectra. I might add some of them later.
Thanks to Charles Rezk for discussion (already a while back).
gave base change geometric morphism its own dedicated paragraph
I added some results and references at Calkin algebra after I noticed that Zoran had added some comments about set-theoretic axioms leading to different properties. In particular the outer automorphism algebra of the Calkin algebra is trivial or not, depending on whether one has CH, or something that violates CH, Todocevic’s Axiom.
Created a page for DLO, the (first-order) theory of . Made some notes about model-theoretic properties, Cantor’s theorem that all countable models of the theory are isomorphic, and also remarked (based on an exercise from Mac Lane and Moerdijk) that the subobject classifier for the topos can be naturally identified with the Dedekind cuts on .
Created Fraïssé limit.
(I was pleasantly surprised to see @David_Corfield had posted about the these things a while ago for the n-Category Cafe.)
Mentioned a neat result of Olivia Caramello’s that omega-categorical structures presentable as Fraisse limits are determined by their automorphism groups with the topology of pointwise convergence in a very nice way: their classifying toposes are precisely the toposes of continuous -sets.
To fill in a grey link, I also created an entry for the countable random graph.
created model structure on dg-comodules, just so as to record a pointer to Positelski 11, theorem 8.2.
Regarding the dg-comodules which are injective as graded comodules over the underlying graded cocommutative co-algebra: Suppose the latter is co-free and the ground ring is a field. Is it then true that all injective comodules over it are cofree? Because this would seem to be a dual version of the Quillen-Suslin theorem?
There is a reflective coequalizer completion, much like the regular and exact completions. The construction seems to be due to Pitts, see e.g. Ch17 of the book on Algebraic Theories. What would be the right way to add this to the nlab? There are probably more references, and the page on regular and exact completions could be improved a little. What is the nPOV?
I changed ‘SEAR is a dependent type theory’ at SEAR to ‘SEAR is a dependently typed theory’. A type theory is a general theory of types, including lots of type formation rules; SEAR is a theory of sets written in a dependently typed first-order logic with very few type formation rules.
But I still linked to dependent type theory, since we don't seem to have good material on using type systems with first-order logic.
created a stub entry for comodule spectrum, for the moment just so as to briefly record the result by Hess-Shipley 14 that comodule spectra over suspension spectra of connected spaces are equivalently parameterized spectra over . Added that reference also to A-theory. Needs to be expanded further.
(Thanks to Charles Rezk for the pointer.)
The stub entry model structure on simplicial Lie algebras used to point to model structure on simplicial algebras. But is it really a special case of the discussion there?
Quillen 69 leaves the definition of the model structure to the reader. Is it with weak equivalences and fibrations those on the underlying simplicial sets? Is this a simplicially enriched model category?
In the references on Infinity-category, I added Emily Riehl’s lecture videos on infinity categories from the Young Topologists’ Meeting 2015.
Added a reference on the smooth manifolds page to -Differentiable Spaces by Navarro Gonzalez and Sancho de Salas, which defines and works with smooth manifolds as locally ringed spaces.
To fill in a grey link (though creating several more in the process), I’ve created Keisler-Shelah isomorphism theorem: two structures have the same theory iff they have an isomorphic ultrapower.
I gave Jones’ theorem (long requested at Hochschild homology) a quick statement and references. Copied this also to the entries free loop space, cyclic loop space and cyclic homology and Sullivan models of free loop spaces:
Let be a simply connected topological space.
The ordinary cohomology of its free loop space is the Hochschild homology of its singular chains :
Moreover the -equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space is the cyclic homology of the singular chains:
(Loday 11)
If the coefficients are rational, and is of finite type then this may be computed by the Sullivan model for free loop spaces, see there the section on Relation to Hochschild homology.
In the special case that the topological space carries the structure of a smooth manifold, then the singular cochains on are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that
This is known as Jones’ theorem (Jones 87)
An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.
created model structure on cosimplicial algebras
but then saw that this should be merged with the previously stubby model structure of cosimplicial rings, so I merged them
for the purposes of having direct links to it, I gave a side-remark at stable Dold-Kan correspondence its own page: rational stable homotopy theory, recording the equivalence
I also added the claim that under this identification and that of classical rational homotopy theory then the derived version of the free-forgetful adjunction
models the stabilization adjunction . But I haven’t type the proof into the entry yet.
stub for model structure on dg-Lie algebras
I gave simplicial Lawvere theory an entry, stating Reedy’s result on the existence of the simplicial model structure of simplicial algebras over a simplicial Lawvere theory
almost missed that meanwhile we have an entry pullback-power. So I added more redirects and expanded a little.
Added the example of smooth manifolds, which have a canonical fully faithful embedding into locally ringed spaces, citing Lucas Braune’s nice proof on stackexchange.
am starting model structure on dg-coalgebras.
In the process I
created a stub for dg-coalgebra
and linked to it from L-infinity algebra
The entry minimal fibration used to be just a link-list for disambiguating the various versions. I have now given it some text in an Idea-section and a pointer to Roig 93 where the concept is considered in generality.
I wanted to understand Borel's Theorem better, so I wrote out a fairly explicit proof of the one-dimensional case.
made curved L-infinity algebra explicit
Started work on syntopogenous space.
I added a remark to inhabited set that one can regard writing to mean “ is inhabited” as a reference to an inequality relation on sets other than denial.
I gave Adams operations some details in the Definition section
I have begun an entry
meant to contain detailed notes, similar in nature to those at Introduction to Stable homotopy theory (but just point-set topology now).
There is a chunk of stuff already in the entry, but it’s just the beginning. I am announcing this here not because there is anything to read yet, but just in case you are watching the logs and are wondering what’s happening. In the course of editing this I am and will be creating plenty of auxiliary entries, such as basic line bundle on the 2-sphere, and others.