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I see Thomas has revived construction in philosophy after it was emptied a while ago from the initial spam. If it’s naming a piece of writing, i.e., by Schelling, we tend to capitilize. But perhaps it’s a more general development within philosophical logic. It would help to relate this entry to constructive mathematics.
In that we’re told its influence continues to our times through Dummett, etc., are we to think of that strand of constructivism which runs through to Martin-Lof? I recently noted him say
…mathematical knowledge through the construction of concepts, Ger. mathematische Erkenntnis durch die Konstucion der Begriffe, a splendid formulation which no doubt had a fruitful influence on Brouwer, and to my mind it is justifiable to say that intuitionism is a development of an essentially Kantian position in the foundations of mathematics. (Martin-Löof, Analytic and synthetic judgements in type theory, p. 99).
After the third time typing "pretopology" into my nlab-goto box and ending up at pretopological space instead of where I wanted to be, namely Grothendieck pretopology, I changed the redirect. It seems likely to me that the latter notion will be of more interest to more of our clientele than the former. But if you disagree, speak up.
I also added a Wikipedia-style "see also" note to the top of Grothendieck pretopology. Should we do that sort of thing in general?
hilbertthm90 has mentioned he wants to write some substance on abelian variety, so to encourage and facilitate this, I have created a short stub for it, mainly the bibliography and short idea.
New stub tangent map.
It uses the link differential of a map which does not direct to anything at the moment as it is hard to decide. The entry differential is dedicated to differential of a chain complex, hence neglecting the term usage for the differential of a map of Banach spaces or the differential of a map of differentiable manifolds. Now the nLab mostly uses derivative for a differential and at th moment derivative points to differentiable map. Now there is an entry differentiation which is covering mostly the same as differentiable map but in the way of Lawvere-Kock synthetic differential geometry, Inside the entry differentiation there is a place whete derivative and differential are contrasted in a way which is exactly opposite to the traditional analysis: the entry calls derivative an infinitesimal difference and differential the ratio, while all classical textbooks do it the opposite to that. Moreover, in that entry, the link differential is used which points to chain complexes, hence nothing to do (Urs was always complaining that the expression derived functor is not motivated although differentials in chain complexes are used to do it), hence we should not mix differentials in homological algebra and differential of a map, which should become a good redirect, once we agree upon conventions or possible mergers of entries. Attention Urs, Todd, Toby, Mike.
There is a stub adjoint representation which in my opinion should be the same entry as adjoint action, hence should be merged. Words representation and action are in general equivalent; to each action one assigns a representation and viceversa (up to nicetess of inner hom spaces etc.). True, the specialists in Lie theory like to prefer calling representation when they have a linear representation but their own textbooks start with nonlinear case. Thus action of a Lie group on a Lie group is nonlinear hance usually the action terminology used while on a Lie algebra more often the representation is used, but it is not a rule, and the distinction does not survive in generalizations (like the Hopf algebra); any sensible entry, as the main entry “adjoint action” should relate th nonlinear case and its linearization hence should not be in separate entries. I wanted to write some references for adjoint action for quantum groups but gave up as I do not know into which of the entries and expect a decision on fiture fate of the two entries first.
I’ve made a few changes to covering lifting property, fixing the reference and adding one to the Elephant. I also added a redirect from comorphism of sites.
Later I might spell out the theorems as numbered environments.
Someone created a page called characteristic variety, but with silly content. It has been renamed empty 160. (The content was TJM-37MOONG BEAN and TJM-37 would seem to be a disease resistant variety of Mung bean!)
edits and edit discussion on the entry conformal compactification is going on here
Started topologically cyclic group, as needed at global family.
I updated separable space. I have two questions:
stub for conserved current
started projective module
(will need to move some material around with projective object. Also, I am splitting off now projective resolution from resolution )
The entry (infinity,1)-Kan extension is still a sad stub which you shouldn’t look at if you have better things to do. But I have now briefly added at least a few more specific pointers to HTT, in particular to the pointwise-ness issue. But just pointers, essentially no text for the moment. (If you feel energetic, be invited to turn the entry into something prettier!)
Added some more information about the properties and possible variations, plus a reference to its properties.
Added a page about a colored generalization of the notion of a symmetric sequence at symmetric colored sequence. I’m happy to merge this (or some heavily edited and corrected version of it) with the page on symmetric sequences. Also open to massive edits or whatever. Just feel like something like this should be on here.
Couldn’t find a latest changes discussion for symmetric sequence so I am just reporting that I added a little bit to that page. In particular, I added another slicker definition in the case that we are interested in a symmetric sequence for the sequence of symmetric groups.
Created exact square, but haven’t linked to it from anywhere else yet. I’m planning to move some of the discussion of exactness at derivator to its own page homotopy exact square, analogous to this one.
(Is the phrase “exact square” used for other things that we should worry about disambiguating/clarifying?)
I added some information to Tractatus Logico-Philosophicus. As this topic (interpretation of the Tractatus) is really complex (and I certainly do not have complete knowledge) and I am new to nLab, I am not quite sure about what and how much to put in. Any advice?
Urs is doing a demo on the nlab. He is elaborating the axiom of determinacy. Set theorist, please elaborate.
started a minimum at E-nilpotent completion (the thing that an -Adams tower converges to).
Made a page for wheeled graph, which follows directly from generalized graph and draws from the recent book of Hackney, Robertson and Yau. Will add more later about how this relates to properads and PROPs, as well as graphical sets, which generalize simplicial sets and dendroidal sets.
Created a new page generalized graph based on the definition given in Hackney, Robertson and Yau’s recent book, which appears to be influenced at least in part by the paper of Kock cited on the page. As far as I can tell none of the other things on the nlab (e.g. quiver or graph or their associated sub-entries about pseudographs and so forth) deal with the case of the “exceptional cell.” If the notion I describe on this page is already somewhere on the nlab, I’d be happy to know that and get rid of the page I made.
I have expanded a good bit the discussion of the classical Adams spectral sequence. In order to simplify the cross-referencing, for the moment I have added the material not yet at classical Adams spectral sequence nor at May spectral sequence but at
But mostly I did expand on details of the May spectral sequence applied to the classical case, for instance I have spelled out some of the computations that go into the identification of the differential on the second page (e.g. here).
Might anyone have a pdf copy of Peter May’s thesis for me, the document that, I gather, expands over the published version May 66 by a discussion specific to the Steenrod algebra?
At Sweedler notation, Zoran wrote
One can formalize in fact which manipulations are allowed with such a reduced notation.
Where can I find such a formalization?
I gave commutative Hopf algebroid its own entry. (There used to be allusions to this concept at Hopf algbroid and at Hopf algebroid over a commutative base).
Besides the definition, I added discussion of the commutative Hopf algebroids arising as generalized dual Steenrod algebras, copied over from the coresponding section at Adams spectral sequence.
stub for relativistic particle (in order to record a good reference kindly pointed out by Igor Khavkine to me)
I created the article coarse structure.
Created the page Fraenkel-Mostowski model and populated it with basic definitions and information moved from ZFA
As I need trees as relational structures for my modal logic stuff, I have added something on them in tree. I took this from of the definition from Blackburn et al, and it seems to me to be related to the previous paragraph in tree. Any thoughts? I am not 100% happy with my addition as there seem to be some awkwardness in their formulation.
I am trying to go towards a n-dim analogue of modal stuff, and here the relation has properties that state that its converse relation is a partial function with domain the set of nodes minus the root. Looking at this homotopically it says that forms part of a contracting homotopy. This in turn suggests that there is a perhaps useful n-relational structure obtained from an n-dim contractible complex (generalising a converse to simple homotopy operations). Has anyone come on something like this explicitly? The idea is a fairly elementary one to have but a nPOV type development might be fun at least.
I have started a page localization of abelian groups with a bunch of facts and their proofs.
Expanded G-set to talk about topological groups as well.
Split off p-adic integer from p-adic number.
I’ve started a page category of G sets. I will continue to fill in more details as I have time. Some of the information from ZFA can probably be moved here.
I created the article motivic coarse spectrum.
I created the article motivic coarse space.
I created the article bornological coarse space.
added the statement that every abelian group admits a free resolution of length 2, here.
The page constant morphism says
As with Set, any morphism which factors through a terminal object is constant but although this is an “if and only if” in Set it need not be in a general category.
I think the two are equivalent in any category with a terminal object, as explained here. That is, a morphism is constant if and only if the natural transformation factors through the terminal object in the presheaf category, so that if the latter is representable then so is the factorization of .
Have I missed something? If not I’ll edit the page.
brief paragraph at Dolbeault cohomology
I have added the definition of the incarnation of EM-spectra as symmetric/orthogonal spectra (here)
Added to Eilenberg-MacLane space a section Cohomology – Of EM spaces with a minimum of pointers and the statement of the rational cohomology ring of .
The recently added link to the stack project pdf file does not work it seems.
I have spelled out the derivation of the Gysin sequence from the Serre spectral sequence at Gysin sequence .
I added to principle of equivalence the following quote:
Either a thing has properties that nothing else has, in which case we can immediately use a description to distinguish it from the others and refer to it; or, on the other hand, there are several things that have the whole set of their properties in common, in which case it is quite impossible to indicate one of them. For if there is nothing to distinguish a thing, I cannot distinguish it, since otherwise it would be distinguished after all. (Tractatus §2.02331)
and a suggestive comment after it about considering such a statement for any given language, rather than the global setting stated in §1 (’the world’).
In the language given by the internal logic of a category one can never distinguish objects that share all their properties! My thesis is that ideas such as the internal language of a category and the Yoneda lemma have precursors in Tractatus, but I’ve not had time to sit down and nut out the details. Others have written about identifying the logic of the Tractatus (eg Potter’s The logic of the Tractatus, Weiss’ Logic in the Tractatus I: Definability, but I haven’t done a thorough search). I haven’t added these comments to the nLab anywhere, but I hope to do so when I flesh out my arguments.
In the process, I added to Tractatus Logico-Philosophicus some online sources for the text.
Created duality involution.
Created 2-category with contravariance and 3-category with contravariance, combining ideas from this preprint of mine with ideas that Vladimir Sotirov had written at contravariant functor (which I moved from there to here).
Created 2-dinatural transformation. The extended example at the bottom is taken from what Vladimir Sotorov wrote at contravariant functor; I moved it over because this is what it’s an example of, but I don’t entirely understand the category-of-enriched-categories part of the example.
Apparently we still didn’t have a page for strict 2-natural transformation, so I made a stub.
Stub for eventual image
I have started a category:reference entry for the article Mandell, May, Schwede, Shipley: Model categories of diagram spectra. I feel that, maybe with the aid of hindsight, there is now room for a somewhat more concise and streamlined presentation of the material in there, for instance by separating basic category theory from the actual constructions a bit more, and I am in the process of typing that into the entry.
So far I worked on part I.
I’ll polish this a bit more, then I am going to feel inclined to copy this over to relevant sections in the entries on sequential-, symmetric- and orthogonal spectra, respectively, for completeness.
There is already a page Ulrik Buchholtz and now we have UlrikBuchholtz.
I have created an entry model structure on topological sequential spectra.
In parts this directly parallels the entry Bousfield-Friedlander model structure.
But now I have spelled out full proof of the model structure and its cofibrant generation: here
I did this by taking the more general proof that I had earlier spelled out at Model categories of diagram spectra, and specializing it to the case of sequential spectra.
The effect of that is that those tedious technical lemmas about the maps of free spectra collapse to something simple, with the result that the actual proof may start right away with less preliminaries, which makes the writeup a bit more transparent. On the other hand, the neat thing is that apart from that analysis of the free spectra the proof is verbatim the same now for all cases (sequential, symmetric, orthogonal spectra and pre-excisive functors), so in the other entries it’ll be possible to turn this around and say: “after this analysis of the free symmetric/orthogonal spectra the proof of their model structure now follows verbatim as at model structure for topological sequential spectra”.
As far as exposition and writeup goes, the only remaining “gap” I left is that at one point the proof invokes that and hence is a topological model structure (this is used in the proof of this lemma ). I plan to spell that out, too. But not tonight.
created The Music of the Spheres, following Ravenel.
Tried to clarify the history and the relationship between the different models at symmetric monoidal smash product of spectra.
expanded/polished a little the Definition at exact couple
At monoidal category the pointers to
#Kelly
in the first two lemmas are broken. There is no item that they point to. What’s the reference?