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- Discussion Type
- discussion topicalgebraic dynamics
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 29th 2012

Changes to dynamical system. Stub for algebraic dynamics.

- Discussion Type
- discussion topicstructure
- Category Latest Changes
- Started by zskoda
- Comments 15
- Last comment by zskoda
- Last Active May 28th 2012

New entry structure but the $n$Lab is down so I save here the final version of editing, which is probably lost in $n$Lab:

The concept of a

*structure*is formulated as the basic object of mathematics in the work of Bourbaki.In model theory, a structure of a language $L$ is the same as model of $L$ with empty set of extra axioms. Given a first-order language $L$, which consists of symbols (variable symbols, constant symbols, function symbols and relation symbols including $\epsilon$) and quantifiers; a

**structure**for $L$, or $L$-structure is a set $M$ with an**interpretation**for symbols:if $R\in L$ is an $n$-ary relation symbol, then its interpretation $R^M\subset M^n$

if $f\in L$ is an $n$-ary function symbol, then $f^M:M^n\to M$ is a function

if $c\in L$ is a constant symbol, then $c^M\in M$

Interpretation for an $L$-structure inductively defines an interpretation for well-formed formulas in $L$. We say that a sentence $\phi\in L$ is true in $M$ if $\phi^M$ is true. Given a theory $(L,T)$, which is a language $L$ together with a given set $T$ of sentences in $L$, the interpretation in a structure $M$ makes those sentences true or false; if all the sentences in $T$ are true in $M$ we say that $M$ is a model of $(L,T)$.

Some special cases include

**algebraic structures**, which is usually defined as a structure for a first order language with equality and $\epsilon$-relation both with the standard interpretation, no other relation symbols and whose function symbols are interpreted as operations of various arity. This is a bit more general than an**algebraic theory**as in the latter, one needs to have free algebras so for example fields do not form an algebraic theory but are the algebraic structures for the theory of fields.In category theory we may talk about functor forgetting structure (formalizing an intuitive, related and in a way more general sense), see

- Discussion Type
- discussion topicalgebraic group
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 5
- Last comment by zskoda
- Last Active May 28th 2012

I added in the definition of algebraic group the requirement ”field” into ”algebraically closed field”. Alternatively one could omit ”field” in the definition at all since this is implicit in ”variety”.

- Discussion Type
- discussion topiccoring
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 28th 2012

New sections at coring:

- base extension of corings
- morphisms in the 1-category of corings over variable base rings.

- Discussion Type
- discussion topicn-angulated category
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 28th 2012

- Discussion Type
- discussion topicCalculus is Topology
- Category Latest Changes
- Started by isomorphisms
- Comments 8
- Last comment by DavidRoberts
- Last Active May 28th 2012

- I recently came across some interesting ideas at inperc.com/wiki/index.php?title=Calculus_is_topology which might be incorporable into the nLab wiki -- although I'm not sure exactly where.

- Discussion Type
- discussion topiclectures on p-divisible groups
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 4
- Last comment by Stephan A Spahn
- Last Active May 27th 2012

I took some notes during my reading of (Chapters I-III of)

- Michel Demazure, lectures on p-divisible groups web

In the recent days I inserted parts of these notes in different nlab entries. Maybe it is of use to somebody to have all of these notes in wiki-form. So I created lectures on p-divisible groups containing the skeleton of the contents. I will fill in the parts I have written so far (roughly chapters I and II) tomorrow. Of course anyone should feel free to rewrite or complete the related entries. Currently the page names contain the chapter-numbering from the original text - I think this numbering can be discarded at the time the linked page contains more information than the original text. If there is some nlab policy on wiki-ed texts suggesting otherwise, please let me know.

- Discussion Type
- discussion topicdependent type formulation of internal diagrams
- Category Latest Changes
- Started by IngoBlechschmidt
- Comments 3
- Last comment by Mike Shulman
- Last Active May 26th 2012

- I've added a formulation of internal diagrams using dependent types.

- Discussion Type
- discussion topicvielbein
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active May 26th 2012

I have added to

*vielbein*a discussion of the vielbein as an orthogonal structure and, more in details as an example of a tiwsted differential c-structure for $\mathbf{c} : \mathbf{B} O(n) \to \mathbf{B} GL(n)$.

- Discussion Type
- discussion topicdefinable set
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 25th 2012

Finally, definable set.

- Discussion Type
- discussion topiclocally bounded category
- Category Latest Changes
- Started by Emily Riehl
- Comments 7
- Last comment by FinnLawler
- Last Active May 25th 2012

I created locally bounded category with basic results from papers of Kelly and Lack. My motivation (unfortunately not reflected in the current stub) is to provide a reference for convergence conditions for the free monad construction.

On this topic, does anyone know whether there are reasonable conditions under which the dual “free comonad” construction would converge? I’m concerned by the result at locally presentable category (new to me; does anyone have a reference?) which says that the opposite of a locally presentable category is locally presentable only if the category is a poset.

- Discussion Type
- discussion topictheory
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 25th 2012

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 $\phi$ in $T$, its interpretation in $M$, $\phi^M$ is true (“$\phi$ 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, $T_1$, $T_2$ 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 $M$.

- Discussion Type
- discussion topicanalytic space
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by zskoda
- Last Active May 24th 2012

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 $\infty$-topos of $\infty$-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.

- Discussion Type
- discussion topicGalois-Lascar group
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active May 23rd 2012

Stub for Lascar group, the analogue of Galois group for first order theories.

- Discussion Type
- discussion topicgeneralized vielbein
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 22nd 2012

created

*generalized vielbein*

- Discussion Type
- discussion topicformal scheme
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 5
- Last comment by Stephan A Spahn
- Last Active May 22nd 2012

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.

- Discussion Type
- discussion topicgenus / Witten genus
- Category Latest Changes
- Started by Urs
- Comments 48
- Last comment by Urs
- Last Active May 22nd 2012

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.

- Discussion Type
- discussion topicWhitehead tower
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active May 21st 2012

I treid to clean up Whitehead tower a bit:

I rewrote and expanded the Idea/Definition part.

Then I

*moved*David Roberts' material that was there to the appropriate section at the new Whitehead tower in an (infinity,1)-topos. (There I tried to add some introductory remarks to it but will try to further highlight David's results here in a moment).At Whitehead tower I left just a new section that says that there is a notion of Whitehead towers in more general contexts with a pointer to Whitehead tower in an (infinity,1)-topos

- Discussion Type
- discussion topicdouble cover
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active May 21st 2012

felt like creating

*double cover*

- Discussion Type
- discussion topicgeneralized tangent bundle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active May 21st 2012

felt like creating

*generalized tangent bundle*

- Discussion Type
- discussion topicactions
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by TobyBartels
- Last Active May 21st 2012

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.

- Discussion Type
- discussion topiclocally cartesian closed category
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Mike Shulman
- Last Active May 21st 2012

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.

- Discussion Type
- discussion topicstack semantics for locally internal categories
- Category Latest Changes
- Started by IngoBlechschmidt
- Comments 2
- Last comment by zskoda
- Last Active May 20th 2012

- I've added a few examples to stack semantics illustrating how it can be used to talk about locally internal categories (in the sense of the appendix of Johnstone's Topos Theory).

- Discussion Type
- discussion topicBarratt-Eccles operad
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by zskoda
- Last Active May 18th 2012

stub for Barratt-Eccles operad

- Discussion Type
- discussion topiccategorical wreath product
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active May 18th 2012

categorical wreath product. Finally.

- Discussion Type
- discussion topicdivisible group
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 4
- Last comment by zskoda
- Last Active May 18th 2012

I added the characterization of a divisible abelian group as an injective object in the category of abelian groups to divisible group.

- Discussion Type
- discussion topicquasicompact
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by TobyBartels
- Last Active May 17th 2012

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.

- Discussion Type
- discussion topicweak equivalence of internal categories
- Category Latest Changes
- Started by DavidRoberts
- Comments 7
- Last comment by Urs
- Last Active May 17th 2012

Started weak equivalence of internal categories. Needs some more work, including examples and theorems.

- Discussion Type
- discussion topicuniformly hyperfinite algebra
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by zskoda
- Last Active May 17th 2012

stub for

*uniformly hyperfinite algebra*

- Discussion Type
- discussion topicGamma spaces and theories
- Category Latest Changes
- Started by fpaugam
- Comments 9
- Last comment by fpaugam
- Last Active May 16th 2012

- I would like to understand what kind of theory Gamma is, in the sense of doctrines.

This is not included on the Gamma-space page, and i would like something about

that to be here, so this discussion is making a proposition in this sense.

I mean that Delta maclane may be seen as the algebraic theory of monoids (category

with finite products opposite to that of finitely generated free monoids), and its

models in sets are monoids, and in categories are monoidal categories (pseudo-functors).

For Gamma, it is not an algebraic theory, since n is not the product of n times 1, but

it seems to me something like a monoidal theory. I would tend to define Gamma of

Segal as a kind of theory of commutative monoids in monoidal categories, obtained

by adding to the operations in Delta (monoidal structure) additional symmetry data.

For example, if i take a monoidal functor from Gamma to sets, i get a commutative

monoid, whose underlying monoid is the model of Delta (algebraic theory), but

the theory Gamma seems monoidal, not algebraic in the sense of Lawvere.

These notions are important to better understand higher categorical generalizations

of monoidal and symmetric monoidal categories.

Am i correct here? May we add these theoretical/doctrinal considerations to the Gamma-spaces

page to clarify where Gamma is coming from, from a conceptual/theoretical point of view?