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• Some of you may remember that a while ago I had started wondering how one could characterize geometric morphisms of toposes $E \to F$ that would exhibit $E$ as an “infinitesimal thickening” of $F$.

Instead of coming to a defnite conclusion on this one, I worked with a concrete example that should be an example of this situation: that of the Gorthendieck toposes on the sites CartSp and ThCartSp of cartesian spaces and infinitesimally thickened cartesian spaces.

But now I went through my proofs for that situation and tried to extract which abstract properties of these sites they actually depend on. Unless I am mixed up, it seems to me now that the essential property is $CartSp$ is a coreflective subcategory of $ThCartSp$ and that in the respective adjunction

$CartSp \stackrel{\leftarrow }{\hookrightarrow} ThCartSp$

buth functors preserve covers.

So maybe it makes sense to take this as a definition: a geometric morphism of Grothendieck toposes is an infinitesimal thickening if it comes from such a coreflective embedding of sites.

Details of this, with more comments on the meaning of it all and detailed proofs, I have now typed into my page on path oo-functors in the section Infinitesimal path oo-groupoids.

• I added a disambiguation note to conjunction, since most of the links to that page actually wanted something else. Then I changed those links to something else: logical conjunction (not yet extant).

An Internet and dictionary search suggests that there is no analogous danger for disjunction (also not yet extant).

• Wrote two-sided bar construction. There is a lot to add, but I added a query box under the subsection “Delooping machines” which I’m hoping someone like Mike could answer.

• I’ve created mate, but I can’t get my nice fancy diagrams to display – I just get the source. Help! What have I done wrong?

• Tim van Beek has written about unbounded posets at partial order.

Where is this used?

• In another thread I came up with a definition of a local isomorphism in a site, working from the definition of a local homeomorphism/diffeomorphism in Top/Diff respectively (with the open cover pretopology in both cases). Then I find that there is a page local isomorphism talking about maps in presheaf categories: such a map is a local isomorphism if becomes an isomorphism on applying the sheafification functor $PSh(S) \to Sh(S,J)$. To quote my definition again

Definition: Let (C,J) be a site (J a pretopology). A map $f:a \to b$ is a J-local isomorphism if there are covering families $(v_i \to b)$ and $(u_j \to a)$ such that for each $u_j$ the restriction $f|u_j$ is an isomorphism onto some $v_i$.

I don’t claim, in the time I have available, to understand the implications of the definition at local isomorphism. I just wonder how it relates to concrete notions like local homeomorphisms (let us work with Top and open covers as covering families). Is a local homeomorphism, after applying Yoneda, a local isomorphism? Does a local isomorphism in the image of Yoneda come from a local homeomorphism? I suspect the answer is yes. Now for the biggie: can a local isomorphism be characterised in terms as basic as my definition as quoted? With my definition one avoids dealing with functor categories (and so size issues, to some extent: $[Top^{op},Set]$ is very big), so if they are equivalent, I’d like to put this somewhere.

Obviously we can take the site in my definition to be a presheaf category with the canonical pretopology or something, and potentially recover the definition at local isomorphism, but for the ease of connecting with geometric ideas, I prefer something simpler.

Any thoughts?

• Urs has erased the sentence explanining the purpose of the entry. Why ??

"In fact not only that it is a good survey but it has a nice bibliography. The main plan of this entry is to build a hyperlinked bibliography of the above article!"

Geometric and topological structures related to M-branes

• Started thinking about smooth paths.

(Incidentally, David, do you want query boxes added to your web? And would you like to change the CSS for off-web links from those boxes to some nice colour?)

• I felt the need to write down what it means for a subspace to have the Baire property, so I did.

• A discussion of the cartesian closed monoidal structure on an (oo,1)-topos is currently missing on the nLab.

I started making a first step in the direction of including it:

• at model structure on simplicial presheaves I added a section Closed monoidal structure with a pointer to Toen’s lectures (where the following is an exercise) and a statement and proof of how $[C^{op},sSet]_{proj}$ is a monoidal model category by the Cartesian product.

• as a lemma for that I added to Quillen bifunctor the statement that on cofib generated model cats a Quillen bifunctor property is checked already on generating cofibrations (here).

More later…

• Based on recent discussions here primarily with Harry and Urs, I added a proof at co-Yoneda lemma in terms of extranaturality, and some didactic material over at adjunction bridging hom-functors to units/counits, involving some but hopefully not too much overlap with related material Urs recently added at adjoint functor. Still need to work in some links.

• I created cylinder on a presheaf and will fill it in more as I read through Ast308. I plan on adding more stuff as I get to it (things about test categories and localisers, etc.).

This is similar but not the same as cylinder object, since even though it is specialized to presheaf categories, we don’t require any notion of a weak equivalence a priori.

• I have quietly submitted the beginning of an article on "surface diagrams" on my web. There is still quite a lot left to write up, and it needs to be formatted more prettily, but I thought I'd throw what I have (so far) out there.

• I have started an entry on pre-Lie algebras, which are much more interesting than you might think at first. My friend Bill Schmitt, the combinatorist, is visiting and telling me amazing things about combinatorics and operads.... this is a little bit of the story.
• I moved the characterization of pointwise kan extensions as those preserved by representable functors to the top (of the section on pointwise kan extensions) and made it the definition (since there was no unified definition before). This is for aesthetic reasons. Since being pointwise is a property, I like that this property has a definition independent of the computational model we’re using.

Are there size issues that I might be glossing over?

• I added a paragraph about $AC_{00}$ to countable choice

• I think the definition of the Grothendieck construction was wrong. The explicit definition was right, but the description in terms of a generalized universal bundle didn’t work out to that, if by “the category of pointed categories” was meant for the functors to preserve the points, which is the usual meaning of a category of pointed objects. I corrected this by using the lax slice. Since while I was writing it I got confused with all the op’s, I decided that the reader might have similar trouble, so I changed it to do the covariant version first and then the contravariant.

• I expanded the Examples-section at petit topos and included a reference to Lawvere’s “Axiomatic cohesion”, which contains some discussion of some aspects of a characterization of “gros” vs “petit” (which I wouldn’t have noticed were it not for a talk by Peter Johnstone).

I am thinking that it should be possible to give more and more formal discussion here, using Lawvere’s article and potentially other articles. But that’s it from me for the time being.

• Swapped the order of the propositions that small limits commute with small limits and that limits commute with right adjoints, which allowed me to give a proof that small limits commute with small limits by citing the result on right adjoints and the characterization of the limit as right adjoint to the constant diagram functor.

• Started the article dependent choice, and did some editing at COSHEP to make clearer to myself the argument that COSHEP + (1 is projective) implies dependent choice. It’s not clear to me that the projectivity of 1 is removable in that argument; maybe it is.

• Started a stub at family of sets. This should also explain concepts like a family of subsets of a given set or a family of groups. And how to formalise them all in material and structural set theories, predicative foundations, internally in indexed categories, etc.

• An anonymous coward put something blank (or possibly some spam that somebody else blanked within half an hour) at Hausdorff dimension, so I put in a stub.

• I moved the proof of the claim that the Segal-Brylinski “differetiable Lie group cohomology” is that computed in the (oo,1)-topos of oo-Lie groupoids from the entry group cohomology to the entry Lie infinity-groupoid and expanded the details of the proof considerably.

See this new section.

Towards the end I could expand still a bit more, but I am not allowed to work anymore today… :-)

• I’ve added a bit about these to free monoid. (These are the computer scientists’ stacks, not the geometers’ stacks!) There is a query about queues too; I’ve forgotten something and can’t reconstruct it.