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Added to reflected limit the example that fully faithful functors reflect limits and colimits (here).
We needed a page negative moment, so I started one.
for completeness: unitization
Since Urs started Freudenthal magic square, I added magic triangle. Does the ’magic pyramid’ of Duff and colleagues in A magic pyramid of supergravities warrant an entry?
Hmm, if the triangle extends the square, and the pyramid has the square as a base, shouldn’t there be a magic tetrahedron?
I have recently created an entry definable set. It is usually defined as an equivalence class of formulas satisfied by the same elements in each structure of the first order language (or model for a theory). But some works recently, like the lectures of David Kazhdan (pdf), take for granted the observation that in fact the relation of equivalence implies that the evaluation of definable sets in a model extends to a functor on the category of models and elementary monomorphismsm, say . There are now many other notions in model theory which have ’definable” as a prefix, and they do not fill uniformly in the same pattern. For example, the category of “definable spaces” and “definable continuous maps” is not the same as the functor from to topological spaces and continuous maps, though possibly some examples would probably fit in the latter description. Also some “definable” notions are for a fixed model and not functors on .
Now, Hrushovski in his 2006 work looked at definable groupoids, about which I just plan to create an entry. Now I was cautious not to say that it is simply an internal groupoid in the category of definable sets, as it is at first defined differently. Besides dealing with internal objects in a large functor category, one should possibly make care about this as well, regarding that we are dealing with a setup where one should be maybe careful about tools like large cardinals (what would be the elegant way to do?).
But in any case, the first problem (that the definition was not given as a functor) can easily be dealt.
So the definition roughly says that one has definable set of objects and of morphisms (just as we wanted), but then the structure maps like target, composition etc. should be definable functions, hence, as relations they are definable subsets of cartesian products, so when presented from as functors into the categories of sets and relations, which are functional in every model. So, now I just checked a very simple fact, specific to the category of sets and functions, that
Proposition. Given a small category and two functors the natural transformations are in 1-1 correspondence with functors such that and is a functional relation.
In other words, functoriality of is the same as being natural. That is new to me.
This immediately implies that the definable groupoid as in Hrushovski arxiv/math.LO/0603413 is (up to the delicacies dealing with functor categories) indeed an internal groupoid in the category of functors and natural transformations.
I finally realized that this ought to exist. And sure enough, it had been constructed already: the 4d supergravity Lie 2-algebra-extension of the 4d super-Poincaré super-Lie algebra. I have added a minimum of an Idea-section and pointers to the references.
I’ve expanded Zoran’s entry on type (in model theory), with an ideas section in the language of categorical logic. I’ve also moved definitions to a definition section and briefly mentioned saturation, monster models, and how Barr-exactness of the syntactic category (i.e. elimination of imaginaries) leads to a model-theoretic account of Galois theory—there’s a grey link there which I will flesh out soon.
Hi everyone!
I’ve created relative adjoint functor, and linked to it from the local definition of adjoint functor (a partially defined adjoint yields an adjoint relative to the inclusion of a full subcategory).
and is as far as I know nonstandard notation, but I think it’s ok, even if the left subscript feels a bit kludgy. I will add more stuff in the next few days.
PS: Thanks a lot to all the nLab contributors; in the past few years I’ve learn a lot through here :) I now have the time and a little bit of confidence to contribute, so any pointers, tips, formatting, style suggestions, whatever will be greatly appreciated
There is a new personal page created with the name Debapriyay, it doesn’t look great. I haven’t got time at the moment to address it.
I had splitt-off quadratic refinement from quadratic form and expanded slightly
started a minimum at de dicto and de re, mainly such as to have a place to point to section 4 of
As promised in this thread, I’ve written up how model theorists treat Galois theory at model-theoretic Galois theory. I briefly mention the connection with the theory of internal covers and the Tannakian formalism. The proof of the fundamental order-reversing bijection between closed subgroups and intermediate extensions is essentially Poizat’s, but I don’t mention the corresponding Galois theory for the strong Stone space (types taken in ), which is the language he prefers.
I’ve also filled out necessary references at monster model, model-theoretic algebraic closure, and theory of algebraically closed fields.
I have expanded a little at CW-spectrum
at universal construction there used to be a little chat between me and Toby along the lines of "would be nice if somebody added something here".
Since I think by now we have plenty of pointers to this entry, I thought it should present itself in a slighly more decent fashion. So I removed our chat and left a stubby but honest entry.
added at exact sequence two small lemmas on forming quotients in an exact sequence
I created a short entry Banach-Alaoglu theorem to fulfil a grey link. While the general theorem is equivalent to the Tychonoff theorem for Hausdorff spaces, the case of separable Banach spaces is constructive. I do wonder how constructive, but apparently this gets used to construct solutions to PDEs, so I guess it’s quite concrete.
created a stub for dihedral homology, for the moment just so as to record a recent reference
Am starting an entry pro-manifold. Have added statement and proof that pro-Cartesian spaces are fully faithful in smooth loci (here).
I have written an “exegesis” of Lawvere’s Some Thoughts on the Future of Category Theory (see that link).
A version of this I have also posted in reply to this MO question
There is a dead link at ionad (the one to ‘web’. This is repeated at Richard Garner. There is a second link, (to an ArXiv version), so I will delete the dead ones. Is there a published version somewhere as well?
There was an old entry faithful representation. I have edited and expanded it a little. While I was was at it, I have added the definition of faithful infinity-actions in an infinity-topos.
(Nothing non-trivial here, just for completeness.)
I made some progress understanding the concept of Albanese variety and updated this page.
The miracle here is that being an abelian variety is just a property of a pointed connected projective algebraic variety, not extra structure.
As Qiaochu Yuan pointed out on MathOverflow, any basepoint-preserving continuous map between tori is homotopic to a group homomorphism. But when these tori are abelian varieties, and the map preserves that structure, it’s actually equal to a group homomorphism!
Expanded the examples section at end (prodded by discussion in the blog), stating Kan extension and geometric realization.
Also added a toc.
There is a strange entry Metadata properties which was created bt ‘The User’ quite some time ago. Does anyone know what this was? (I happened on it when searching for something else.)
I’ve added a proof that rank 0 virtual vector bundles are nilpotent elements in K-theory to virtual vector bundle, and made some small edits to topological K-theory. The latter page had the definition of K-theory mixed up with the definition of reduced K-theory, so I made a small correction there, but the proof that is a ring is not there now, as the existing proof was for reduced K-theory only. I will edit this in the coming days.
Incidentally, for ’compact support’ vector bundle K-theory of a non-compact space (so not representable K-theory), this implies that the -adic topology is discrete, as opposed to the case where for non-compact spaces representable K-theory has something interesting going on (cf work of Atiyah-Segal on equivariant K-theory and completion, where they compute in the case of representable K-theory for compact).
created an entry for caloron correspondence with an Idea-section and references.
Someone calling themselves‘Realigned arrows for clarification and fixed composition order’ has changed the composition order on hom-functor.
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