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• In need a definition of an action of a groupoid object $G$ in an ($\infty$,1)-category (actually in an ($\infty$,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.

• 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!).

• 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.

• 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.

• 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 $X^\sharp$ where Lurie uses $X^\sharp$ and $X^\natural$. That seems gratuitously confusing to me; is there a reason for it?

• I have created a stub HNN-extension. I have been wondering how to link in the connection between homotopy colimits and graphs of groups (see Fiore, Luck and Sauer), any ideas? Perhpas it will have to wait until there is a graphs of groups and a complexes of groups entry

• 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 $X \to X/E$, 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 $C \leftarrow X \rightarrow D$ of full, faithful, essentially surjective functors. Or equivalently, a pair $C \leftrightarrow D$ 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.

• I have added details to product type on both positive and negative definitions, with the corresponding beta and eta reduction rules.

• The statement at compact support was that $f^{-1}(0)$ should be compact. I’ve corrected this.

• 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!)

• I put an actual definition at theorem. It is still quite the stub, however!

• I am going to polish the entry local system now.

The following is long forgotten discussion that had been sitting in a query box there. Everybody involved should check what of that still needs further discussion and then have that discussion here on the forum.

Urs: I am hoping that maybe David Speyer, whose expositional blog entry is linked to below, or maybe somebody else would enjoy filling in some material here.

Bruce: Could it perhaps be “On a topological space (why do we need connected?) this is the same as a sheaf of flat sections of a finite-dimensional vector bundle equipped with flat connection;”. I guess by “flat connection” in this general topological context we would mean simply a functor from the homotopy groupoid to the category of vector spaces?

Zoran Škoda: connected because otherwise you do not have even the same dimension of the typical stalk of teh lcoally constant sheaf. Maybe there is a fancy wording with groupoids avoiding this, but when you have a representation on a single space, you need connectedness.

Ronnie Brown I do not have time to write more tonight but mention that there is a section of the paper

• (with P.J.HIGGINS), “The classifying space of a crossed complex”, Math. Proc. Camb. Phil. Soc. 110 (1991) 95–120.

on local systems, where a module over the fundamental groupoid of a space is regarded as a special case of a crossed complex. This seems convenient for the singular theories but has not been developed in a Cech setting. The homotopy classification theorem

$[X, \mathcal{B}C] \cong [\Pi X_* ,C]$

where $X_*$ is the skeletal filtration of the CW-complex $X$, $C$ is a crossed complex, and $\mathcal{B}C$ is the classifying space of $C$, thus includes the local coefficient version of the classical Eilenberg-Mac Lane theory.

Tim: Quoting an exercise in Spanier (1966) on page 58:

A local system on a space $X$ is a covariant functor from the fundamental groupoid of $X$ to some category.

A reference is given to a paper by Steenrod: Homology with local coefficients, Annals 44 (1943) pp. 610 - 627.

Perhaps the entry could reflect the origins of the idea. The current one seems to me to be much too restrictive. There are other applications of the idea than the ones at present indicated, although of course those are important at the moment. Reference to vector bundles is not on the horizon in Spanier!!!!.

Local systems with other codomains than vector spaces are used in rational homotopy theory.

Urs: I am all in favor of emphasizing that “local system” is nothing but a functor from a fundamental groupoid. That’s of course right up my alley, compare the discussion with David Ben-Zvi at the “Journal Club”. Whoever finds the time to write something along these lines here should do so (and in clude in particular the reference Ronnie Brown gives above).

BUT at the same time it seems to me that many practitioners will by defualt think of the explicitly sheaf-theoretic notion when hearing “local syetem” which the entry currently states. We should emphasize this explicitly, something like: “while in general a local system is to be thought of as a representation of a fundamental groupoid, often the term is understood exclusively in its realization within abelian sheaf theory as follows …”

(to be continued in next comment)

• Let $\mathcal{Z}$ be the Zariski topos, in the sense of the classifying topos for local rings. I was wondering whether there might be any connection between $\mathbf{Sh}(\operatorname{Spec} \mathbb{Z})$ and $\mathcal{Z}$. Certainly, there is a geometric morphism $\mathcal{Z} \to \mathbf{Sh}(\operatorname{Spec} \mathbb{Z})$, and there’s also a geometric inclusion $\mathbf{Sh}(\operatorname{Spec} \mathbb{Z}) \to \mathcal{Z}$. On the other hand, there’s no chance of $\mathcal{Z}$ itself being localic, since it has a proper class of (isomorphism classes of) points. Let’s write $L \mathcal{Z}$ for the localic reflection of $\mathcal{Z}$; the first geometric morphism I mentioned then corresponds to a locale map $L \mathcal{Z} \to \operatorname{Spec} \mathbb{Z}$. But what is $L \mathcal{Z}$ itself?

The open objects in $\mathcal{Z}$ can be identified with certain saturated cosieves on $\mathcal{Z}$ in the category of finitely-presented commutative rings, and so may be identified with certain sets of isomorphism classes of finitely-presented commutative rings. If I’m not mistaken, every finitely-presented commutative ring gives rise to an open object in $\mathcal{Z}$. This suggests that $L \mathcal{Z}$ might be some kind of (non-spatial) union of all isomorphism classes of affine schemes of finite type over $\mathbb{Z}$, which is an incredibly mind-boggling thing to think about. It’s not clear to me whether other kinds of open objects exist. For example, does every not-necessarily-affine open subset of $\operatorname{Spec} A$, for every finitely-presented ring $A$, also show up…?

• I’ve inserted some proofs of statements made at Heyting algebra, particular on the “regular element” left adjoint to the full inclusion $Bool \to Heyt$.

The proof that $L \to L_{\neg\neg}$ preserves implication seemed harder than I was expecting it to be. Or maybe my proof is a clumsy one? If anyone knows a shorter route to this result, I’d be interested.

• by chance I noticed that two days ago somebody created an entry Circuitoids. I am not sure what to do about it…

• at 2-adjunction I would like to list a bunch of 2-category theoretic analogs of standard facts about ordinary adjunctions. Such as: a right adjoint is a full and faithful 2-functor precisely if the counit of the 2-adjunction is an equivalence, etc.

But I haven’t really thought deeply about 2-adjunctions myself yet. Is there some reference where we could take such a list of properties from?

• I wanted to add some stuff about completely distributive lattices, when I got annoyed by the fact that few of the entries on lattices, frames, etc, carried a table of contents, and that I kept being surprised by which related entries already existed and which not.

Just a start. Please feel free to expand.

• I've made some small additions to the article on algebraic lattices (including fixing the languishing typo in the introduction, "An alegbraic lattice is...").
• I stumbled across a nice reference while looking for something else, so added it to axiom of choice so I might read it all later.

• I have added a little bit to spine.

(Will maybe write out the proof of the proposition there in a little while.)

• I have worked a bit on 2-congruence.

The main addition is that I started an Examples-section, where I started writing out an explicit proof (little exercise in unwinding the definitions) of the statement:

The 2-category of 2-congruences in $Grpd$ is equivalent to that of small categories.

One should write out more. But it is getting late for me now. I should continue another day.

• Added more material to Boolean algebra, particularly the principle of duality and the connection to Boolean rings, and a wee bit of material on Stone duality.

Stone duality deserves greater expansion, bringing out the dualities via ambimorphic (ahem, schizophrenic) structures on the 2-element set, and mentioning the connection to Chu spaces. Another day, another dollar.

• it had annoyed me for a long time that we had now dedicated entry for Giraud theorem. I have now created one, so that the redirects no longer simply go to Grothendieck topos. But then, I didn’t have the energy to add more than a few pointers, for the moment.

• I noticed a link to page for atomic Boolean algebra , so I made the page. It's pretty simple. People may wish to add examples or something.
• I have created a stub for Bob Thomason as there were frequent references to the Thomason model category structure in the nLab but no link to an entry on the person and his work. The St. Andrews biography is good, giving good details on his work. I have linked to it but perhaps we need some more comment in the nLab entry.

• I added some random meanings to order, then noted that they’re not entirely unrelated, since these orders can be ordered.

• added to Kan lift the def of absolute Kan lifts, and some more examples.

Btw, in all of the sources I’ve read about this, Kan lifts are simply called (left, right) liftings. What do you think about renaming this to liftings? will it conflict with some other kind of “lifting”?

• I have added more to higher generation by subgroups. As I said on another thread, this material feels as if there should be a nPOV / categorified version that could be quite interesting, so any thoughts would be welcome.

• New entry universal epimorphism redirectinig also universal monomorphism. It is not among those variants listed in epimorphism. We also do not list absolute epimorphism (epimorphism which stays epimorphism after applying any functor to it). Every split epimorphism stays split after applying a functor hence it is absolute, but is there a counterexample of an absolute epimorphism which is not in fact split ?

By the way, here is an archived version of the old query from strict epimorphism

David Roberts: I’m interested in a bicategorical version of this. You haven’t happened to have done this Mike?

Mike Shulman: Not more than can be extracted from 2-congruence (michaelshulman) and regular 2-category (michaelshulman). What is there called an “eso” is the bicategorical version of a strong epi (which agrees with an extremal epi in the presence of pullbacks), and what is there called “the quotient of a 2-congruence” is the bicategorical version of a regular epi. I’ve never thought about the bicategorical version of a strict epi; since strict epis agree with regular epis in the presence of finite limits I’ve never really had occasion to care about them independently.

• A graduate student at Johns Hopkins who is being supervised by Jack Morava, named Jon(athan) Beardsley, wrote a short article Bousfield Lattice. More on this in a moment.

• I created locally regular category and added a corresponding section to allegory.

Edit: removed some complaints that were due to it being too late at night and my brain not working correctly.

• I am experimenting with a notion of Heisenberg Lie $n$-algebras, for all $n \in \mathbb{N}$.

I have made an experimental note on this here in the entry Heisenberg Lie algebra.

It’s explicitly marked as “experimental”. If it turns out to be a bad idea, I’ll remove it again. Please try to shoot it down to see if I can rescue it! :-)

I mean, the definition in itself is elementary and very simple. The question is if this is “the right notion” to consider. The reasoning here is:

by the arguments as mentioned on the nCafé here we may feel sure that Chris Rogers’s notion of Poisson Lie n-algebra is correct. (Not that there were any particular doubts, but the fact that we can derive it from very general abstract homotopy theoretic constructions reinforces belief in it.)

But the ordinary Heisenberg Lie algebra is just the sub-Lie algebra of the Poisson Lie algebra on the constant and the linear functions. Therefore it makes sense to look at the sub-Lie $n$-algebra of the Poisson Lie $n$-alhebra on the constant and linear differential forms That’s what my experimental definition does.

• Added some relevant bits to connected limit, fiber product, and pushout. I wanted to record the result at connected limit that functors preserve connected limits iff they preserve wide pullbacks, which may be a slightly surprising result if one has never seen it before.