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

• brief category:people-entry for hyperlinking references

• I was a bit confused about a search for “pullback lemma” not returning any result, hence this redirect

• Zimo Sun, A note on the representations of $SO(1,d+1)$ (arXiv:2111.04591)
• 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.

• I do not quite understand the discussion under the axiom of extensionality. It says that one can do two variants (of course), one is that equality is taken as a primitive (predicate) and another that the axiom is taken as a definition of equality. Of course, this is in the idea so, but then the second version of the axiom is written unclear (and it is not quite a definition) and the discussion is also unclear. I see it this way: there is a propositional calculus and there is a propositional calculus with equality. “The same” is probably here just an informal way to refer to the equality of the propositional calculus with equality, that means a distinguished binary predicate which satisfies substitution axiom, transitivity, reflexivity and symmetry. In that calculus we just state the axiom (not a definition) that the equality can be rephrased via belongness relation. If we work in just a propositional calculus then we can define a relation which we may call equality but we can also call it blablabla. Then we have to state explicitly which properties, possibly weaker than required in the propositional calculus with equality hold for that relation (I am not quite sure how entry extensional relation solves this, in particular, should we now prove the substitution axiom for equality). Now, one should do that version precisely. In calculus with equality there is no space for two versions as far as I see: equality is given (distinguished) by the propositional calculus with equality, we can only characterize it.

• I had added to Practical Foundations a pointer to the accompanying Foundations of computable topology.

Any chance that somebody has an electronic copy of the Practical Foundations which he or she could borrow me for second?

• I have added to coequalizer basic statements about its relation to pushouts.

In the course of this I brought the whole entry into better shape.

• I made the former entry "fibered category" instead a redirect to Grothendieck fibration. It didn't contain any addition information and was just mixing up links. I also made category fibered in groupoids redirect to Grothendieck fibration

I also edited the "Idea"-section at Grothendieck fibration slightly.

That big query box there ought to be eventually removed, and the important information established in the discussion filled into a proper subsection in its own right.

• I added the definition of a filtered (infinity,1)-category from HTT. Since this is performed in a simplicial model which is supposedly not to be emphasized from the nPov and I felt that the below proposition should center this article I added a sentence indicating this in the ”Idea”.

• starting something. There is nothing to be seen yet, but I need to save.

• Added a new Properties section to connected object. Including a theorem which is a bit of a hack (where I leave it to others to decide if ’hack’ should be interpreted positively or negatively!).

• Updated the webpage link. He has been at Lille for some time.

• This is intended to continue the issues discussed in the Lafforgue thread!

I have added an idea section to Morita equivalence where I sketch what I perceive to be the overarching pattern stressing in particular the two completion processes involved. I worked with ’hyphens’ there but judging from a look in Street’s quantum group book the pattern can be spelled out exactly at a bicategorical level.

I might occasionally add further material on the Morita theory for algebraic theories where especially the book by Adamek-Rosicky-Vitale (pdf-draft) contains a general 2-categorical theorem for algebraic theories.

Another thing that always intrigued me is the connection with shape theory where there is a result from Betti that the endomorphism module involved in ring Morita theory occurs as the shape category of a ring morphism in the sense of Bourn-Cordier. Another thing worth mentioning on the page is that the Cauchy completion of a ring in the enriched sense is actually its cat of modules (this is in Borceux-Dejean) - this brings out the parallel between Morita for cats and rings.

• Created:

## Definition

Denote by $Emb_n$ the site of $n$-dimensional smooth manifolds and open embeddings.

An (∞,1)-sheaf $F\colon Emb_n^op\to Top$ of topological spaces is microflexible if for any closed inclusion $K\to K'$ of compact spaces, the induced map $F(K')\to F(K)$ is a Serre microfibration.

An (∞,1)-sheaf $F\colon Emb_n^op\to Top$ of topological spaces is flexible if for any closed inclusion $K\to K'$ of compact spaces, the induced map $F(K')\to F(K)$ is a Serre fibration.

## Gromov’s theorem

Given an open manifold $M$, the inclusion of microflexible sheaves into flexible sheaves on the site $Emb_n/M$ is an equivalence of (∞,1)-categories.

## Related concepts

• the standard bar complex of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.

I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).

(That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)

• I have given free category its own little entry. This used to be (and still is) the topic of one subsection at path category, but I feel that for pointing people to just the plain concept of a free category, the entry “path category” is not focused enough. But both entries remain cross-linked, so the reader may still explore further, if desired.

• Added a general result by Gromov.

• The bare minimum, to satisfy links.

• Created:

## Definition

A map $p$ of topological spaces is a Serre microfibration if for any lifting square for $\{0\}\times K\to [0,1]\times K$ and $p$, we can find $\epsilon>0$ such that the lifting property is satisfied after restricting to $[0,\epsilon]\times K\subset [0,1]\times K$.

## Properties

Any Serre fibration is a Serre microfibration.

Any inclusion of open subspaces is a Serre microfibration. It is a Serre fibration if and only it is a homeomorphism.

## Related concepts

• Added a remark amplifying that the 0-simplex really has no horn, and that one must not think it could be defined to be the empty set (saw long and unresolved MO discussion of this point…)

• Following the discussion here, I have cleared this page, and removed the three pointers to it, which were at balanced category, at isomorphisms and at bireflective subcategory (all of them doing nothing but claiming the terminology).

A fourth mentioning of the word (without a link) was and is at power-associative algebra, where it is used to mean binary multimorphism (and again, that page wouldn’t lose anything if it just said that instead of “bimorphism”).

• starting a stub. Nothing here yet, but need to save.

• brief category:people-entry for hyperlinking references

• more hyperlinks (and some whitespace) to the paragraph on maximal tori.

• the statement that smooth actions of compact Lie groups on smooth manifolds are proper

• tried to bring the entry Lie group a bit into shape: added plenty of sections and cross links to other nLab material. But there is still much that deserves to be done.

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

• some minimum, in order to make the link work

• some minimum

• starting something. Not much here yet besides the definition, but need to save.

• am finally giving this an entry

• brief category:people-entry for hyperlinking references

• I added two characterisations of weak homotopy equivalences to model structure on simplicial sets.

For the record, I found the inductive characterisation in Cisinski’s book [Les préfaisceaux comme modèles des types d’homotopie, Corollaire 2.1.20], but I feel like I’ve seen something like it elsewhere. The characterisation in terms of internal homs comes from Joyal and Tierney [Notes on simplicial homotopy theory], but they take it as a definition.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• I made some minor improvements to the Properties section of pushout, making it match the similar section in pullback insofar as it can. (It’s a bit tiring to have to look at both these pages to get all the basic properties, so I fixed that, but for properties that hold both for pullbacks and dually for pushouts I’m happy to have all the proofs at pullback - that’s how it works now.)

• I have created a page for Turing categories linked to the already-existing page on restriction categories, with some sections for near-future expansion. First I will add some references, since there is by now a small industry devoted to this framework for categorical recursion theory, and it forms the basis of a lot of other ongoing work, for example in differential categories and higher realizability models.

• Fixed some hickups in the very first sentence.

• while bringing some more structure into the section-outline at comma category I noticed the following old discussion there, which hereby I am moving from there to here:

[begin forwarded discussion]

+–{.query} It's a very natural notation, as it generalises the notation $(x,y)$ (or $[x,y]$ as is now more common) for a hom-set. But personally, I like $(f \rightarrow g)$ (or $(f \searrow g)$ if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from $f$ to $g$. —Toby Bartels

Mike: Perhaps. I never write $(x,y)$ for a hom-set, only $A(x,y)$ or $hom_A(x,y)$ where $A$ is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen $[x,y]$ for an internal-hom object in a closed monoidal category, and for a hom-set in a homotopy category, but not for a hom-set in an arbitrary category.

I would be okay with calling the comma category (or more generally the comma object) $E(f,g)$ or $hom_E(f,g)$ if you are considering it as a discrete fibration from $A$ to $B$. But if you are considering it as a category in its own right, I think that such notation is confusing. I don’t mind the arrow notations, but I prefer $(f/g)$ as less visually distracting, and evidently a generalization of the common notation $C/x$ for a slice category.

Toby: Well, I never stick ‘$E$’ in there unless necessary to avoid ambiguity. I agree that the slice-generalising notation is also good. I'll use it too, but I edited the text to not denigrate the hom-set generalising notation so much.

Mike: The main reason I don’t like unadorned $(f,g)$ for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see $(f,g)$ in a category is that we have $f:X\to A$ and $g:X\to B$ and we’re talking about the pair $(f,g):X\to A\times B$ — surely also a natural generalization of the very well-established notation for ordered pairs.

Toby: The notation $(f/g/h)$ for a double comma object makes me like $(f \to g \to h)$ even more!

Mike: I’d rather avoid using $\to$ in the name of an object; talking about projections $p:(f\to g)\to A$ looks a good deal more confusing to me than $p:(f/g)\to A$.

Toby: I can handle that, but after thinking about it more, I've realised that the arrow doesn't really work. If $f, g: A \to B$, then $f \to g$ ought to be the set of transformations between them. (Or $f \Rightarrow g$, but you can't keep that decoration up.)

Mike: Let me summarize this discussion so far, and try to get some other people into it. So far the only argument I have heard in favor of the notation $(f,g)$ is that it generalizes a notation for hom-sets. In my experience that notation for hom-sets is rare-to-nonexistent, nor do I like it as a notation for hom-sets: for one thing it doesn’t indicate the category in question, and for another it looks like an ordered pair. The notation $(f,g)$ for a comma category also looks like an ordered pair, which it isn’t. I also don’t think that a comma category is very much like a hom-set; it happens to be a hom-set when the domains of $f$ and $g$ are the point, but in general it seems to me that a more natural notion of hom-set between functors is a set of natural transformations. It’s really the fibers of the comma category, considered as a fibration from $C$ to $D$, that are hom-sets. Finally, I don’t think the notation $(f,g)$ scales well to double comma objects; we could write $(f,g,h)$ but it is now even less like a hom-set.

Urs: to be frank, I used it without thinking much about it. Which of the other two is your favorite? By the way, Kashiwara-Schapira use $M[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$. Maybe $comma[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$? Lengthy, but at least unambiguous. Or maybe ${}_f {E^I}_g$?

Zoran Skoda: $(f/g)$ or $(f\downarrow g)$ are the only two standard notations nowdays, I think the original $(f,g)$ which was done for typographical reasons in archaic period is abandonded by the LaTeX era. $(f/g)$ is more popular among practical mathematicians, and special cases, like when $g = id_D$) and $(f\downarrow g)$ among category experts…other possibilities for notation should be avoided I think.

Urs: sounds good. I’ll try to stick to $(f/g)$ then.

Mike: There are many category theorists who write $(f/g)$, including (in my experience) most Australians. I prefer $(f/g)$ myself, although I occasionally write $(f\downarrow g)$ if I’m talking to someone who I worry might be confused by $(f/g)$.

Urs: recently in a talk when an over-category appeared as $C/a$ somebody in the audience asked: “What’s that quotient?”. But $(C/a)$ already looks different. And of course the proper $(Id_C/const_a)$ even more so.

Anyway, that just to say: i like $(f/g)$, find it less cumbersome than $(f\downarrow g)$ and apologize for having written $(f,g)$ so often.

Toby: I find $(f \downarrow g)$ more self explanatory, but $(f/g)$ is cool. $(f,g)$ was reasonable, but we now have better options.

=–

• Below the statement that the right class is stable under $\infty$-limits in the arrow category, I added the statement (here) that it is in fact reflective in the arrow category.

[edit: ah, sorry, only now do I see that this was already stated in the page, but in another subsection – will merge… ]

• In checking for how to best link to this statement from within a proof, I realized that, apparently, there was no real statement recorded anywhere on the $n$Lab.

So here a bare minimum, the bare statement with pointer to HTT, just so that I can link to it.

• This entry had a section “Abuses” (originating in rev 1) which contained exactly the following paragraph:

It is often said that two diagrams are cofinal even when neither has a colimit, if they acquire a common colimit on passing to a suitable completion of $C$. This can probably be phrased internally to $C$, at the cost of intuition.

I am hereby removing this section for the time being. Maybe there is something to be said here, but the headline “Abuses” seems disconnected from what the paragraph says, the first sentence remains vague and the second and last sentence seems to call into question whether the author of these lines had thought this through.

If anyone knows what useful statement the author of these lines had in mind, let’s sort it out, phrase it clearly, and then add it back to the entry.

• Mentioned that fully faithful morphisms can (should) be defined with respect to a proarrow equipment.

• started a Properties-section at Lawvere theory with some basic propositions.

Would be thankful if some experts looked over this.

Also added the example of the theory of sets. (A longer list of examples would be good!) And added the canonical reference.