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• Started this page having heard people speak about Tambara modules.

• Andre Joyal just created on the nLab an entry titled CatLab

• brief category:people-entry for hyperlinking references

• Created:

Not to be confused with Arthur Lewis Stone.

Arthur Harold Stone was a general topologist. He got his PhD degree in 1941 from Princeton University, advised by Solomon Lefschetz.

## Related entries

• We have several entries that used to mention Lawvere’s fixed point theorem without linking to it. I have now created a brief entry with citations and linked to it from relevant entries.

• I added to and revised the Idea section of diagonal argument, and added some references.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• Added a comment on comonads as oplax 2-functors.

Some of this (condition 2 in the proposition in the section on algebras) was written as a preparatory step for a to-be-written nLab article on Day’s reflection theorem for symmetric monoidal closed categories, which came up in email with Harry and Ross Street.

• brief category:people-entry for hyperlinking references

• Several recent updates to literature at philosophy, the latest being

• Mikhail Gromov, Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2., pdf; Structures, Learning and Ergosystems: Chapters 1-4, 6 (2011) pdf

which is more into cognition and language problem, but still very relevant, and by a top mathematician. As these 2 are still manuscripts I put them under articles, though I should eventually classify those as books…

• For some $n$Lab work that I will do in the next days, I want to be able, for convenience, to refer the reader to an entry which exhibits the iterated lifting-property-calculus that is nicely discussed in the appendix of the article by Joyal-Tierney on complete Segal spaces:

For the moment I titled that entry Joyal-Tierney calculus. Not sure, though, if that is a good name. Suggestions would be welcome.

• I added a reference in the section on terminology to Makkai’s ’Towards a categorical foundation of mathematics’, where he defines what he calls the ’Principle of Isomorphism’. This is essentially what ’evil’ captures, I think, and it is handy to have a published version with a sensible name to which to refer people.

Here’s a wild thought: what about renaming the page principle of isomorphism and having evil redirect there. It would necessitate a rewrite of the page, but still contain material about the jokey names (evil, kosher etc). I recall that someone here told how some of these in-jokes are off-putting to outsiders or newcomers (Zoran, maybe?). Just an idea.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• added to equivariant K-theory comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.

(Also finally added references to Green and Julg at Green-Julg theorem).

This all deserves to be prettified further, but I have to quit now.

• Page created, but author did not leave any comments.

Anonymous

• co-page of the “bi-initial object” page just created

Antonin Delpeuch

• better reference welcome! And I hope I did not insert any mistake by reformulating.

Antonin Delpeuch

• I have made explicit (here) that a large but accessible and tensored $\infty$-category is still a full subcategory of its prop-objects, in the expected way.

• added a cross reference to prorepresentable functor

The link to Marc Hoyois notes is dead. Perhaps his higher Galois theory has much the same content (as I do not know the notes refered to here and his webpage does not seem to list this pdf file.)

• Created Ho(Cat), mainly as a place to put a counterexample showing that it doesn't have pullbacks. If anyone has a simpler one, please contribute it.

• am starting some minimum here. Have been trying to read up on this topic. This will likely become huge towards beginning of next year

• seeing Eric create diffeology I became annoyed by the poor state that the entry diffeological space was in. So I spent some minutes expanding and editing it. Still far from perfect, but a step in the right direction, I think.

(One day I should add details on how the various sites in use are equivalent to using CartSp)

• Someone anonymous has changed the wording in coherent module. I think it is correct but it is a bit incoherent. They changed ’Noetherian ring’ to ’coherent ring’ but have not defined that concept. Can someone who knows the terminology better than me check this out? It probably just needs a line saying what a coherent ring is.

• a stub, for the moment just for completeness, in order to make links work

• Added more lowbrow, quicker-to-understand definition of “coherent ring”.

• I have added in references to Whitehead’s address ’delivered before the Princeton Meeting of the AM Society on November 2, 1946’ that is ‘combinatorial homotopy 1’.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• Page created, but author did not leave any comments.

• moving the following old discussion from out of the entry to here, just for the record (it concerns a bygone version of the entry):

+– {: .query} Tim: As I read the entry on nice topological spaces, it really refers to ’nice categories’ rather than ’nice spaces’! I have always thought of spaces such as CW-complexes and polyhedra as being ’locally nice’, but the corresponding categories are certainly not ’nice’ in the sense of nice topological space. Perhaps we need to adjust that other entry in some way.

Toby: You're right, I think I've been linking that page wrongly. (I just now did it again on homotopy type!) Perhaps we should write locally nice space or locally nice topological space (you pick), and I'll fix all of the links tomorrow.

Tim:I suggest locally nice space. (For some time I worked in Shape Theory where local singularities were allowed so the spaces were not locally nice!) There would need to be an entry on locally nice. I suggets various meanings are discussed briefly, e.g. locally contractible, locally Euclidean, … and so on, but each with a minimum on it as the real stuff is in CW-complex etc and these are the ’ideas’.

Mike: Why not change the page nice topological space to be about CW-complexes and so on, and move the existing material there to something like convenient category of spaces, which is also a historically valid term? I am probably to blame for the current misleading content of nice topological space and I’d be happy to have this changed.

Toby: I thought that nice topological space was supposed to be about special kinds of spaces, such as locally compact Hausdorff spaces, whose full subcategories of $\Sp$ are also nice. (Sort of a counterpoint to the dichotomy between nice objects and nice categories, whose theme is better fit by the example of locally Euclidean spaces). CW-complexes also apply —if you're interested in the homotopy categories.

Mike: Well, that’s not what I thought. (-: I don’t really know any type of space that is nice and whose corresponding subcategory of Top is also nice. The category of locally compact Hausdorff spaces, for instance, is not really all that nice. In fact, I can’t think of anything particularly good about it. I don’t even see any reason for it to be complete or cocomplete!

I think it would be better, and less confusing, to have separate pages for “nice spaces” and “nice categories of spaces,” or whatever we call them. And, as I said, I don’t see any need to invent a new term like “locally nice.”

When algebraic topologists (and, by extension, people talking about $\infty$-groupoids) say “nice space” they usually mean either (1) an object of some convenient category of spaces, or (2) a CW-complex-like space, between which weak homotopy equivalences are homotopy equivalences. Actually, there is a precise term for the latter sort: an m-cofibrant space, aka a space of the (non-weak) homotopy type of a CW complex.

Toby: I thought the full subcategory of locally compact Hausdorff spaces was cartesian closed? Maybe not, and it's not mentioned above.

But you can see that most of the examples above list nice properties of their full subcategories. And the page begins by talking about what a lousy category $\Top$ is. So it seems clearly wrong that you can't make $\Top$ a nicer category by taking a full subcategory of nice spaces. (Not all of the examples are subcategories, of course.)

Mike: It’s true that locally compact Hausdorff spaces are exponentiable in $Top$. However, I don’t think there’s any reason why the exponential should again be locally compact Hausdorff.

I guess you are right that one could argue that compactly generated spaces themselves are “nice,” although I think the main reason they are important is that the category of compactly generated spaces is nice. I propose the following:

1. Move the current content of this page to convenient category of spaces.
2. Create m-cofibrant space (I’ll do that in a minute).
3. Update most links to point to one or the other of the above, since I think that in most places one or the other of them is what is meant.
4. At nice topological space, list many niceness properties of topological spaces. Some of them, like compact generation, will also produce a convenient category of spaces; others, like CW complexes, will be in particular m-cofibrant; and yet others, like locally contractible spacees, will do neither.

Toby: I believe that the compact Hausdorff reflection (the Stone–Čech compactification) of $Y^X$ is an exponential object.

Anyway, your plan sounds fine, although nice category of spaces might be another title. (I guess that it's up to whoever gets around to writing it first.) Although I'm not sure that people really mean m-cofibrant spaces when they speak of nice topological spaces when doing homotopy theory; how do we know that they aren't referring to CW-complexes? (which is what I always assumed that I meant).

Mike: I guess nice category of spaces would fit better with the existing cumbersomely-named dichotomy between nice objects and nice categories. I should have said that when people say “nice topological space” as a means of not having to worry about weak homotopy equivalences, they might as well mean (or maybe even “should” mean) m-cofibrant space. If people do mean CW-complex for some more precise reason (such as wanting to induct up the cells), then they can say “CW complex” instead.

Re: exponentials, the Stone-Čech compactification of $Y^X$ will (as long as $Y^X$ isn’t already compact) have more points than $Y^X$; but by the isomorphism $Hom(1,Y^X)\cong Hom(X,Y)$, points of an exponential space have to be in bijection with continuous maps $X\to Y$.

Toby: OK, I'll have to check how exactly they use the category of locally compact Hausdorff spaces. (One way is to get compactly generated spaces, of course, but I thought that there was more to it than that.) But anyway, I'm happy with your plan and will help you carry it out.

=–

• started a Properties-section, with material copied from other entries:

1. the rational cohomology of iterated loop spaces of spheres

2. the relation to configuration spaces of points

• Created.

• added pointer to:

• I gave regular cardinal its own page.

Because I am envisioning readers who know the basic concept of a cardinal, but might forget what “regular” means when they learn, say, about locally representable category. Formerly the Lab would just have pointed them to a long entry cardinal on cardinals in general, where the one-line definition they would be looking for was hidden somewhere. Now instead the link goes to a page where the definition is the first sentence.

Looks better to me, but let me know what you think.

• Page created, but author did not leave any comments.

Anonymous

• I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.

• Page created, but author did not leave any comments.

• added reference to HTT

Shane Kelly

• A query box has been added:

I suspect there is a variant of the definition involving a transformation $R^Z_{X Y} \colon [X,Y] \to [[Y,Z],[X,Z]]$ rather than $L$. Is this correct? If so, how do these two definitions relate? Can one of them be expressed in terms of the other? Or is there a refined definition which comprises both $L$ and $R$?

• brief category:people-entry for hyperlinking references

• Unstub the page with a basic but original blurb. Compare and contrast my article list of complexity classes on esowiki, which has a very different audience.

• starting a bare references-list entry, to be !includeed in the References-sections of relevant entries

Not done yet…