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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• at DHR superselection theory I have added the argument (here) for why every DHR representation indeed comes from a net-endomorphism, assuming Haag duality and that the net takes values in vN algebras.

• 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.

• 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) $\infty$-ring module $\infty$-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.

• I added a characterization, reference, and some more examples to absolute colimit.

• 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 $\kappa$-compact objects in $(\infty,1)Cat$/$\infty Grpd$ are the essentially $\kappa$-small $(\infty,1)$-categories/groupoids.

• I would like to rearrange Kan complexes as ∞-groupoids to something like

1. general description

2. 2-dimensional example

In particular I think the word oriental should occur more prominently in the beginning of this section.

• 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 $\mathbb{Z}_2$-orientation – can be a bit of a pain.

• 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!

• 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.