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- Discussion Type
- discussion topicJohn D. Berman
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by David_Corfield
- Last Active Nov 9th 2021

brief

`category:people`

-entry for hyperlinking references at*equivariant stable homotopy theory*and*enriched (infinity,1)-category theory*

- Discussion Type
- discussion topicPierre Grillet
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_Porter
- Last Active Nov 9th 2021

- Discussion Type
- discussion topicpasting law for pullbacks
- Category Latest Changes
- Started by Tobias Fritz
- Comments 4
- Last comment by Urs
- Last Active Nov 9th 2021

- Discussion Type
- discussion topicde Sitter group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 9th 2021

added pointer to today’s

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

- Zimo Sun,

- Discussion Type
- discussion topiccompactly generated topological space
- Category Latest Changes
- Started by Todd_Trimble
- Comments 80
- Last comment by Urs
- Last Active Nov 9th 2021

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.

- Discussion Type
- discussion topicregular epimorphism
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 9th 2021

added pointer to:

- Jiří Adámek, Horst Herrlich, George Strecker, Def. 7.71 in:
*Abstract and Concrete Categories – The Joy of Cats*John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories**17**(2006) 1-507 (tac:tr17, book webpage, pdf)

- Jiří Adámek, Horst Herrlich, George Strecker, Def. 7.71 in:

- Discussion Type
- discussion topicaxiom of extensionality
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Nov 8th 2021

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.

- Discussion Type
- discussion topicPractical Foundations of Mathematics
- Category Latest Changes
- Started by Urs
- Comments 44
- Last comment by Urs
- Last Active Nov 8th 2021

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?

- Discussion Type
- discussion topiccoequalizer
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Nov 8th 2021

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.

- Discussion Type
- discussion topicGrothendieck fibration
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by varkor
- Last Active Nov 8th 2021

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.

- Discussion Type
- discussion topicfiltered (infinity,1)-category
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 4
- Last comment by nLab edit announcer
- Last Active Nov 8th 2021

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”.

- Discussion Type
- discussion topicslice theorem
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Nov 8th 2021

- Discussion Type
- discussion topicconnected object
- Category Latest Changes
- Started by Todd_Trimble
- Comments 5
- Last comment by Urs
- Last Active Nov 7th 2021

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!).

- Discussion Type
- discussion topicGoncalo Tabuada
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Urs
- Last Active Nov 7th 2021

- Discussion Type
- discussion topicIvo Dell'Ambrogio
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Tim_Porter
- Last Active Nov 7th 2021

- Discussion Type
- discussion topicMorita equivalence
- Category Latest Changes
- Started by Thomas Holder
- Comments 13
- Last comment by Tim_Porter
- Last Active Nov 7th 2021

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.

- Discussion Type
- discussion topicmicroflexible sheaf
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 9
- Last comment by Tim_Porter
- Last Active Nov 6th 2021

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

- Discussion Type
- discussion topichomotopy coherent nerve
- Category Latest Changes
- Started by Urs
- Comments 34
- Last comment by Tim_Porter
- Last Active Nov 6th 2021

added to homotopy coherent nerve two diagrams in the section Examples and illustrations that are supposed to illustrate the hom-SSets of the simplicial category on

- Discussion Type
- discussion topicbar construction
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Tim_Porter
- Last Active Nov 6th 2021

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. ;-)

- Discussion Type
- discussion topiccanonical resolution
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by Tim_Porter
- Last Active Nov 6th 2021

Updated a smidgen to cross-reference with simplicial resolution.

- Discussion Type
- discussion topicfree category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_Porter
- Last Active Nov 6th 2021

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.

- Discussion Type
- discussion topichomotopy coherent diagram
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Tim_Porter
- Last Active Nov 6th 2021

I have added to homotopy coherent diagram and to model structure on algebras over an operad the fact that one can describe homotopy coherent diagrams as algebras over the Boardman-Vogt resolution of some operad. Vogt’s rectification theorem is then a special case of the general Berger-Moerdijk result.

In the course of this I reorganized and expanded homtopy coherent diagram a bit. It still needs to be polished a bit.

- Discussion Type
- discussion topicsimplicial resolution
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Nov 6th 2021

- Discussion Type
- discussion topich-principle
- Category Latest Changes
- Started by David_Corfield
- Comments 5
- Last comment by Urs
- Last Active Nov 6th 2021

- Discussion Type
- discussion topicopen manifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 6th 2021

- Discussion Type
- discussion topicc-principle
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Nov 5th 2021

Created:

## Idea

A variant of h-principle with cobordisms thrown in.

## Related concepts

## References

- Michael Freedman,
*Controlled Mather-Thurston theorems*, arXiv:2006.00374.

- Michael Freedman,

- Discussion Type
- discussion topicSerre microfibration
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Nov 5th 2021

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

- Discussion Type
- discussion topiccone
- Category Latest Changes
- Started by Eric
- Comments 12
- Last comment by Hurkyl
- Last Active Nov 5th 2021

Added a diagram to cone and changed some notation to be compatible with cone morphism and Understanding Constructions in Set

- Discussion Type
- discussion topichorn
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Hurkyl
- Last Active Nov 5th 2021

- Discussion Type
- discussion topicbimorphism > history
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 5th 2021

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”).

- Discussion Type
- discussion topicrelation between quasi-categories and simplicial categories
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Hurkyl
- Last Active Nov 5th 2021

one more remark at relation between quasi-categories and simplicial categories

(to be expanded...)

- Discussion Type
- discussion topicfuzzy sphere
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Nov 5th 2021

- Discussion Type
- discussion topicstraightening functor
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by Urs
- Last Active Nov 5th 2021

Added:

## Related concepts

## References

The original result is due to Lurie:

A considerably simplified presentation is available in

- Fabian Hebestreit, Gijs Heuts, Jaco Ruit,
*A short proof of the straightening theorem*, arXiv:2111.00069.

- Fabian Hebestreit, Gijs Heuts, Jaco Ruit,

- Discussion Type
- discussion topicBrian Williams
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 5th 2021

- Discussion Type
- discussion topiccompact Lie group
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Nov 4th 2021

- Discussion Type
- discussion topicLie group
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Nov 4th 2021

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.

- Discussion Type
- discussion topicJaco Ruit
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Urs
- Last Active Nov 4th 2021

- Discussion Type
- discussion topicglobal quotient orbifold
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 4th 2021

- Discussion Type
- discussion topicpresentable orbifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 4th 2021

- Discussion Type
- discussion topicgood orbifold
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 4th 2021

- Discussion Type
- discussion topicprecision experiment
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Nov 4th 2021

- Discussion Type
- discussion topicLuca Giorgetti
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 4th 2021

- Discussion Type
- discussion topicCartesian fibration
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Dmitri Pavlov
- Last Active Nov 3rd 2021

added more theorems to Cartesian fibration and polished the intro slightly

- Discussion Type
- discussion topicmodel structure on simplicial sets
- Category Latest Changes
- Started by Zhen Lin
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 3rd 2021

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*.

- Discussion Type
- discussion topicLeopold Zoller
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 3rd 2021

- Discussion Type
- discussion topicOliver Goertsches
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 3rd 2021

- Discussion Type
- discussion topicinfinity-category
- Category Latest Changes
- Started by adeelkh
- Comments 6
- Last comment by Urs
- Last Active Nov 3rd 2021

I just added a link to Lurie's "What is...?" paper.

- Discussion Type
- discussion topicpushout
- Category Latest Changes
- Started by John Baez
- Comments 8
- Last comment by Urs
- Last Active Nov 3rd 2021

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.)

- Discussion Type
- discussion topicTuring category
- Category Latest Changes
- Started by James Francese
- Comments 8
- Last comment by Urs
- Last Active Nov 3rd 2021

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.

- Discussion Type
- discussion topicdifferential K-theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Nov 3rd 2021

I have expanded the list of references at

*differential K-theory*with a few comments thrown in.

- Discussion Type
- discussion topicfree action
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Nov 3rd 2021

- Discussion Type
- discussion topiccomma category
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Tim Campion
- Last Active Nov 2nd 2021

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.=–

- Discussion Type
- discussion topicorthogonal factorization system in an (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 2nd 2021

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… ]

- Discussion Type
- discussion topicatlas
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 2nd 2021

I have removed the few stub lines that were in this entry before, and wrote out a more informative Idea-section. Currently it reads as follows:

In basic topology and differential geometry, by an

*atlas*of/for a topological-, differentiable- or smooth manifold $X$ one means a collection of coordinate charts $U_i \subset X$ which form an open cover of $X$.If one considers here the disjoint union $\mathcal{U} \coloneqq \underset{i}{\sqcup} U_i$ of all the choordinate charts, then the separate chart embeddings $U_i \subset X$ give rise to a single map (continuous/differentiable function)

$\mathcal{U} \longrightarrow X$and now the condition for an atlas is that this is a surjective étale map/local diffeomorphism.

If, next, one regards this morphism, under the Yoneda embedding, inside the topos of formal smooth sets, then these conditions on an atlas say that this morphism is

In this abstract form the concept of an atlas generalizes to any cohesive higher geometry (KS 17, Def. 3.3, Wellen 18, Def 4.13).

Next, for a geometric stack $\mathcal{X}$, an atlas is a smooth manifold $\mathcal{U}$ (for differentiable stacks) or scheme $\mathcal{U}$ (for algebraic stacks) or similar, equipped with a morphism

$\mathcal{U} \longrightarrow \mathcal{X}$that is an effective epimorphism and formally étale morphism in the corresponding higher topos (for instance in that of formal smooth infinity-groupoids).

Here the terminology has a bifurcation:

In the general context of geometric stacks one typically drops the second condition and calls any effective epimorphism from a smooth manifold or scheme to a differentiable stack or algebraic stack, respectively, an

*atlas*.If in addition the condition is imposed that such an effective epimorphism exists which is also formally étale, then the geometric stack is called an

*orbifold*or*Deligne-Mumford stack*(often with various further conditions imposed).

- Discussion Type
- discussion topicgroupoid objects in an (∞,1)-topos are effective
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 2nd 2021

- Discussion Type
- discussion topicfunctorial field theory
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Nov 2nd 2021

I am beginning to give the entry

*FQFT*a comprehensive*Exposition and Introduction*section.So far I have filled some genuine content into the first subsection

*Quantum mechanics in Schrödinger picture*.But I have to quit now. This isn’t even proof-read yet. So don’t look at it unless you feel more in editing-mood than in pure-reading-mood.

- Discussion Type
- discussion topiccofinal diagrams
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 2nd 2021

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.

- Discussion Type
- discussion topicfully faithful morphism
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active Nov 1st 2021

- Discussion Type
- discussion topicinfinitary Lawvere theory
- Category Latest Changes
- Started by David_Corfield
- Comments 3
- Last comment by varkor
- Last Active Nov 1st 2021

Added this reference

- Martin Brandenburg,
*Large limit sketches and topological space objects*(arXiv:2106.11115)

- Martin Brandenburg,

- Discussion Type
- discussion topicLawvere theory
- Category Latest Changes
- Started by Urs
- Comments 48
- Last comment by DavidRoberts
- Last Active Nov 1st 2021

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.