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I have expanded theory adding more basics in classical syntactic approach. I added a new subsection
Set-theoretic models for a first-order theory in syntactic approach
The basic concept is of a structure for a first-order language L: a set M together with an interpretation of L in M. A theory is specified by a language and a set of sentences in L. An L-structure M is a model of T if for every sentence ϕ in T, its interpretation in M, ϕM is true (“ϕ holds in M”). We say that T is consistent or satisfiable (relative to the universe in which we do model theory) if there exist at least one model for T (in our universe). Two theories, T1, T2 are said to be equivalent if they have the same models.
Given a class K of structures for L, there is a theory Th(K) consisting of all sentences in L which hold in every structure from K. Two structures M and N are elementary equivalent (sometimes written by equality M=N, sometimes said “elementarily equivalent”) if Th(M)=Th(N), i.e. if they satisfy the same sentences in L. Any set of sentences which is equivalent to Th(K) is called a set of axioms of K. A theory is said to be finitely axiomatizable if there exist a finite set of axioms for K.
A theory is said to be complete if it is equivalent to Th(M) for some structure M.
I have started an entry analytic space with material on Berkovich’s non-rigid analytic geometry.
I don’t really know this subject and have been adding material to the entry as I read about it and to the extent that I correctly understood it. Experts are most welcome to help out.
As indicated here, I am motivated by the following: Berkovich’s local contractibility result suggests that the ∞-topos of ∞-sheaves over the site of p-adic analytic spaces might be cohesive.
The idea would be that his result implies (if it does) that the site (category with coverage) of contractible p-adic afine spaces is a dense subsite of that of all p-adic spaces. Since it should be an infinity-cohesive site that would imply the claim.
But despite looking through Berkovich’s writings for a little bit today, I am still not sure if he just shows that the underlying topological space of a p-adic anayltic space is locally contractible, or if one may indeed deduce that they are locally contractible with respect to étale homotopy, as would be needed for the above conclusion.
Stub for Lascar group, the analogue of Galois group for first order theories.
created generalized vielbein
I added a definition-section to formal scheme with the four equivalent definitions of a k-formal scheme from Demazure, lectures on p-divisible groups. There is some overlap with the section on Noetherian formal schemes now.
created stubs for
This is not supposed to be satisfactory content. I just wanted these pages to exist right now, so that links to them work.
felt like creating double cover
felt like creating generalized tangent bundle
Uday has added :
Mac Lane, VII.4, only requires a monoidal category to define actions. – at action.
This takes up an old point that I made but never felt up to following up.
at locally cartesian closed category I have added a Properties-section Equivalent characterizations with details on how the slice-wise internal hom and the dependent product determine each other.
This is intentionally written in, supposedly pedagogical, great detail, since I need it for certain discussion purposes. But looking back at it now, if you say it is too much notational detail, I will understand that :-). But I think it’s still readable.
stub for Barratt-Eccles operad
I added the characterization of a divisible abelian group as an injective object in the category of abelian groups to divisible group.
I changed quasicompact to quasicompact morphism though it is also about quasicompact schemes etc. as before and moved the query box here:
Mike: To accord with terminological conventions, this page should probably be either “quasicompact space” or “quasicompact object.”
Zoran Skoda: I do not know what are the conventions, but it was intentional to look both at quasicompact spaces and quasicompact morphisms (which are according to the dominant point of view in algebraic geometry, more important and basic notion); and aside also for q. objects. Personally I do not understand English-language preference for noun phrases. If one is to choose, quasicompact morphism is the choice.
Toby: By the «Each definition gets its own page.» convention, I'm not even sure that this shouldn't just redirect to compact space or compact object. My impression is that assuming that ’compact’ implies Hausdorff is either (like assuming that ’ring’ implies commutative) restricted to fields where it's a common assumption or to languages (I'm thinking mostly of Bourbaki in French here) other than English. On the other hand, if it's used that way by English-writing algebraic geometers, then I would seem to be wrong (since algebraic geometers often have non-Hausdorff spaces).
Zoran Skoda: Convention that ’compact’ includes Hausdorff is very common also among people working predominantly on nice spaces, particularly differetial geometers, differential topologists, people studying metric spaces and so on. But for “paracompact” the situation is more tricky: in literature, even on general topology there are also competing definitions, which are all equivalent for Hausdorff spaces. All my life I bounce in such people; my own education does not assume Hausdorffness, unless it is said in the form “compactum”. Algebraic geometers always say quasi-compact, it has nothing to do with language; but as I say for algebraic geometers the basic notion is quasi-compact. The emphasis of this entry is on the terminology and morphisms (what should be expanded on: I still did not write the deifnitions of quasi-compact MORPHISM in various setups); so redirection won’t work I think. Plus although from my point of view saying quasicompact and compact is the same for spaces; one would never say compact for the scheme; scheme is said to be quasicompact if its underlying space is (quasi)compact.
There is an additional reason for that: one can consider a nonsingular variety over complexes which is quasicompact, and which itself is not compact in complex topology (under GAGA). But in the same considerations it is often useful to have some arguments in Zariski and some in complex topology; one of the reasons for word quasicompact is that sometimes we have the “same” example which we are used to think as of noncompact space but it is (quasi)compact in Zariski topology. When an algebraic geometer thinks of the difference between compact and quasicompact for complex varieties he has that in mind; in more general setups about Hausdorff vs nonHausdorff. In the same time, when talking about objects in derived categories of qcoh sheaves, even algebaric geometers use moreoften term compact than quasicompact; thus redirecting to compact object and saying this is for algebraic geometry won’t do for all the 3 notions in this entry (on the contrary side, nobody says compact morphism as far as I could confirm, but quasicompact morphism).Toby: Ah, so when you've got both Zariski and complex topologies around, you can easily distinguish the former by the prefix ’quasi’; that's cute. Anyway, perhaps we'll move this to quasicompact morphism if you write mostly about that, but I won't try to move anything for now.
stub for uniformly hyperfinite algebra
While the lab is down, I’ll collect some stuff here, also to discuss it.
So I am trying to identify in the literature a precise and coherent statement of the supposed adjunction / partial equivalence between “type theories” and “categories”.
In section 8.4.C of Practical Foundations is announced the following, which would be part of that statement:
Unfortunately the online version of the book breaks off right after this announcement, and I don’t have the paper version available at the moment.
Also, that section 8.4.C starts with the word “Conversely”. But where is the converse statement, actually?
I have created a stub field net to go with net of local observables (for the moment mainly such as to record references)
started stub for quantum lattice system, for the moment mainly as a reminder for me concerning the book by Bratteli now referenced there.
As reported elsewhere, Zhen Lin began recursion. I changed the section title “In classical mathematics” to “In general” since there didn’t seem to be anything inherently classical about it. But maybe I’m missing something.
I am working on prettifying the entry contractible type and noticed that where in Categorical semantics it says “Let … with sufficient structure…” we really eventually need to point to an entry that discusses this sufficient structure in detail.
In lack of a better idea, I named that entry presentation of homotopy type theory. Feel free to make better suggestions.
I have tidied up the entry initial algebra and then made sure that it is cross-linked with inductive type (which it wasn’t!).
We really need to rename this entry to initial algebra for an endofunctor. But since I would have to fight the cache bug if I did it now, I decide not to be responsible for that at the moment.
More at field with one element, after creating person entry Christophe Soulé about the creator. By the way the Soulé has different encoding in n-Forum than in nlab so the link does not access the right page from here. See redirect Christophe Soule.
I have added various basic technical details to filtered colimit and flat functor.
created stub for separated geometric morphism
There is room to go through the Lab and interlink all the various entries on separated schemes, Hausdorff spaces etc. pp. and explain how these are all examples of a single notion. But I don’t have the energy for it right now.
New entry Jouanolou cover (prompted by its use in Van den Bergh’s version of a proof that every projective variety is a quiver Grassmanian, which JOhn posts about in cafe). Let me mention also the earlier entry Jean-Pierre Jouanolou.
started (infinity,1)-vector bundle with a bit of discussion of the Ando-Blumberg-Gepner-Hopkins-Rezk theory of (discrete) ∞-ring module ∞-bundles.
I have created exhaustive category — not just the page, but the terminology. No one at MO seemed to know a name for this exactness property, so I made one up. The adjective “exhaustive” seems harmonious with “extensive” and “adhesive”, and expresses the idea that the subobjects in a transfinite union “exhaust” the colimit. But I would welcome other opinions and suggestions.
Created adhesive category.
of course there is also the notion going by the name strongly compact topological space.
started curved dg-algebra
I added a definition to epipresheaf. I am wondering if there is a ”minus construction” turning a presheaf into an epipresheaf.
James Wallbridge put on the arXiv a paper derived from his thesis. I’ve linked to both from his page here. Urs, in particular, was interested in seeing a copy
at compact object in an (infinity,1)-category I have added the definition and stated the examples: the κ-compact objects in (∞,1)Cat/∞Grpd are the essentially κ-small (∞,1)-categories/groupoids.
I would like to rearrange Kan complexes as ∞-groupoids to something like
general description
2-dimensional example
In particular I think the word oriental should occur more prominently in the beginning of this section.
New page: line integral (also redirects from contour integral). Too damn long; somebody should edit this down.
added an illustrating diagram to inverse limit, just so that one sees at one glance what the variance of the arrows is, since following through the “directed/codirected”-terminology and entries – if one really is in need of the ℤ2-orientation – can be a bit of a pain.
In need a definition of an action of a groupoid object G in an (∞,1)-category (actually in an (∞,1)-topos) on an object X - so I created one but I’m not yet sure if it coincides with the existing one if X is pointed.
New stub ergodic theory wanted at measure theory.
started teleparallel gravity
stub for Weitzenböck connection
I have started a new entry on complexes of groups, the higher dimensional version of graphs of groups (in the bass-Serre theory). These are related to orbifolds and topological stacks, but as yet there is just a stub. I have put some stuff in the Menagerie so will transfer more across in a short while (I hope!).
started Eilenberg subcomplex
Have been adding material to model structure for dendroidal complete Segal spaces.
I added to Reedy model structure in the section Over the simplex category a bunch of basic useful lemmas and proofs. It works up to a proof of the Bousfield-Kan map.
for a seminar that we will be running I need a dedicated entry
So I created it.
I inserted a disclaimer on top that there are variants to what people understand under “derived geometry” and point the reader to the entry higher geometry for more details. I would be grateful if we could keep this entry titled this way and discuss variants elsewhere.
I would also be grateful if anyone who feels like making non-controversial edits (typos, references, etc. ) to for the moment do them not on this nLab page, but on this page here on my personal web:
Because currently the content of both pages is identical – except that the latter also has a seminar schedule which is omitted in the former – and until the entry has stabilized a bit more I would like to make edits just in one place and update the other one by copy-and-paste.
quick note local fibration
created reduced simplicial set, just for completeness
Although there is a standard meaning of ‘finite’ in constructive mathematics, it’s helpful to have a way to indicate that one really means this and is not just sloppily writing ‘finite’ in a situation where it is correct classically, without having to make a circumlocution like ‘finite (even in constructive mathematics)’. Based on Mike’s notation at finite set and drawing an analogy with ‘K-finite’, I’ve invented the term ‘F-finite’. (So now the circumlocution is simply ‘finite (F-finite)’ or ‘finite (F-finite)’, assuming that one wishes to relegate constructivism to parenthetical remarks.)
I’ve added this to finite set, added redirects, and used the new abbreviated circumlocution at dual vector space.
I created T. Streicher - a model of type theory in simplicial sets - a brief introduction to Voevodsky’ s homotopy type theory with a summary of that article and linked it from homotopy type theory. Maybe this article can serve as a base for some pedagogical nlab-entry providing some technical details concerning this simplicial model which are omitted in homotopy type theory.
Started entries on Jim Lambek and Phil Scott. These are stubby.
Added to deformation retract the general definition. Moved the previous content to a section Examples - In topological spaces.
It seems that the page marked simplicial set uses X# and X♯ where Lurie uses X♯ and X♮. That seems gratuitously confusing to me; is there a reason for it?
Moving to here some very old discussions from preorder:
Todd says: It’s not clear to me how one avoids the axiom of choice. For example, any equivalence relation E on a set X defines a preorder whose posetal reflection is the quotient X→X/E, and it seems to me you need to split that quotient to get the equivalence between the preorder and the poset.
Toby says: In the absence of the axiom of choice, the correct definition of an equivalence of categories C and D is a span C←X→D of full, faithful, essentially surjective functors. Or equivalently, a pair C↔D of anafunctors (with the usual natural transformations making them inverses).
Todd says: Thanks, Toby. So if I understand you aright, the notion of equivalence you have in mind here is not the one used at the top of the entry equivalence, but is more subtle. May I suggest amplifying a little on the above, to point readers to the intended definition, since this point could be confusing to those inexperienced in these matters?
Urs says: as indicated at anafunctor an equivalence in terms of anafunctors can be understood as a span representing an isomorphism in the homotopy category of Cat induced by the folk model structure on Cat.
Toby says: I think that this should go on equivalence, so I'll make sure that it's there. People that don't know what ’equivalence’ means without choice should look there.
Mike: Wait a minute; I see why every preorder is equivalent to a poset without choice, but I don’t see how to show that every preorder has a skeleton without choice. So unless I’m missing something, the statement that every preorder is equivalent to a poset isn’t, in the absence of choice, a special case of categories having skeletons.
Toby: Given the definition there that a skeleton must be a subcategory (not merely any equivalent skeletal category), that depends on what subcategory means, doesn't it? If a subcategory can be any category equipped with a pseudomonic functor and if functor means anafunctor in choice-free category theory, then it is still true. On the other hand, since we decided not to formally define ’subcategory’, we really shouldn't use it to define ’skeleton’ (or anything else), in which case ‹equivalent skeletal category› is the guaranteed non-evil option. You still need choice to define a skeleton of an arbitrary category, but not of a proset.
Mike: We decided not to formally define a non-evil version of “subcategory,” but subcategory currently is defined to mean the evil version. However, I see that you edited skeleton to allow any equivalent skeletal category, and I can’t really argue that that is a more reasonable definition in the absence of choice.
The thread Category theory vs order theory quickly really became Topological spaces vs locales, so I’m putting this in a new thread.
At category theory vs order theory, I had originally put in the analogy with category : poset :: strict category : proset. Mike changed this to to category : proset :: skeletal category : poset. I disagree. A proset has two notions of equivalence: the equality of the underlying set, and the symmetrisation of the order relation; a poset has only one. Similarly, a strict category has two notions of equivalence: the equality of the set of objects, and the isomorphism relation; a category has only one. I’m OK with using skeletal categories to compare with posets, since this will make sense to people who only know the evil notion of strict category, but I insist on using strict categories to compare with prosets. So now its strict category : proset :: skeletal category : poset.
Just a stub, for the record: Gaussian number.
The statement at compact support was that f−1(0) should be compact. I’ve corrected this.
I expanded the discussion at principal infinity-bundle to go along with the discussion with Mike over at the blog
Urs, I noted you started a new entry on Thomas Hale. Can you check Hale(s) name as his website gives it with an s on the end? I do not know of him so hesitate to change it. (homepages on university websites are not unknown to get things wrong!)