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• 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

!redirects structures

• 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”.

• New sections at coring:

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

• 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.

• 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 $M$.

• 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.

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

• 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.

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.

• 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).
• 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.

• 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?