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- Discussion Type
- discussion topicpretopos
- Category Latest Changes
- Started by Mike Shulman
- Comments 16
- Last comment by Mike Shulman
- Last Active Jul 27th 2020

I have incorporated Jonas’ comment into the text at pretopos, changing the definition to “a category that is both exact and extensive”, as this is sufficient to imply that it is both regular and coherent.

- Discussion Type
- discussion topicWilliam Dunbar
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 27th 2020

brief

`category: people`

-entry for hyperlinking references at*Riemannian orbifold*

- Discussion Type
- discussion topiccategory
- Category Latest Changes
- Started by Mike Shulman
- Comments 12
- Last comment by DavidRoberts
- Last Active Jul 27th 2020

- Discussion Type
- discussion topiccomplete small category
- Category Latest Changes
- Started by Mike Shulman
- Comments 13
- Last comment by Thomas Holder
- Last Active Jul 26th 2020

Created complete small category, and moved the proof of Freyd’s theorem to there from adjoint functor theorem.

- Discussion Type
- discussion topicKan extension
- Category Latest Changes
- Started by Mike Shulman
- Comments 52
- Last comment by Urs
- Last Active Jul 26th 2020

I think the line between the two types of Kan extension (weak versus pointwise) is drawn at the wrong place. Am I missing something?

- Discussion Type
- discussion topicopetopic omega-category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Ali Caglayan
- Last Active Jul 26th 2020

for completeness (prompted by opetopic type theory) I started an entry

*opetopic omega-category*.For me presently this just serves to purpose to record Thorsten Palm’s definition of opetopic omega-category, as I understand it from what Eric Finster tells me.

For the definitions by Baez-Dolan and by Makkai the entry presently only contains placeholders, please feel invited to fill in detail.

All these definitions consider opetopic sets. The difference is in which structure and property is put on that. The original definition of universal cells is somewhat involved, as far as I see. Palm’s definition is of a nice straightforward homotopy-theoretic flavor. It seems plausible that this definition satisfies the homotopy hypothesis, but I don’t know if anyone looked into it.

Accoring to Eric Finster, Palm showed that his definition is a special case of Makkai’s, but the converse remains open.

- Discussion Type
- discussion topicstack
- Category Latest Changes
- Started by zskoda
- Comments 19
- Last comment by varkor
- Last Active Jul 25th 2020

Stack entry says: "The notion of stack is the one-step vertical categorification of a sheaf." In Grothendieck's main works, like pursuing stacks and in the following works of French schools, stack is any-times categorification of a sheaf, and the one-step case is called more specifically 1-stack. We can talk thus about stack in narrow sense or 1-stacks and stacks in wider sense as n-stacks for all n. Topos literature mainly means that the stack is the same as internal 1-stack.

- Discussion Type
- discussion topicVitalyR
- Category Latest Changes
- Started by VitalyR
- Comments 3
- Last comment by DavidRoberts
- Last Active Jul 24th 2020

- Discussion Type
- discussion topicmetric space
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by Mike Shulman
- Last Active Jul 24th 2020

I’ve removed this query box from metric space and incorporated its information into the text:

Mike: Perhaps it would be more accurate to say that the symmetry axiom gives us enriched $\dagger$-categories?

*Toby*: Yeah, that could work. I was thinking of arguing that it makes sense to enrich groupoids in any monoidal poset, cartesian or otherwise, since we can write down the operations and all equations are trivial in a poset. But maybe it makes more sense to call those enriched $\dagger$-categories.

- Discussion Type
- discussion topicindexed category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jul 24th 2020

- Discussion Type
- discussion topicmathematics presented in homotopy type theory
- Category Latest Changes
- Started by David_Corfield
- Comments 9
- Last comment by nLab edit announcer
- Last Active Jul 24th 2020

- Discussion Type
- discussion topicalgebra over a monad
- Category Latest Changes
- Started by Richard Williamson
- Comments 9
- Last comment by Mike Shulman
- Last Active Jul 24th 2020

- Discussion Type
- discussion topicmoduli stack of elliptic curves
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by David_Corfield
- Last Active Jul 24th 2020

I have expanded the Idea-section at

*moduli stack of elliptic curves*, have tried to add more pertinent references, and have touched the subsection on “Over general rings” and on the derived version.In the course of this I started to split off some entries such as

*nodal cubic curve*(which now has a little bit of content) and*cuspidal cubic curve*(which does not yet).

- Discussion Type
- discussion topicBen Moon
- Category Latest Changes
- Started by bgm
- Comments 2
- Last comment by DavidRoberts
- Last Active Jul 23rd 2020

- Discussion Type
- discussion topicTeichmüller theory
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by DavidRoberts
- Last Active Jul 23rd 2020

I have added to

*Teichmüller theory*a mini-paragraph Complex structure on Teichmüller space with a minimum of pointers to the issue of constructing a complex structure on Teichmüller space itself.Maybe somebody has an idea on the following: The Teichmüller orbifold itself should have a neat general abstract construction as the full subobject on the étale maps of the mapping stack formed in smooth $\infty$-groupoids/smooth $\infty$-stacks into the Haefliger stack for complex manifolds : via Carchedi 12, pages 37-38.

Might we have a refinement of this kind of construction that would produce the Teichmüller orbifold directly as on objects in $\infty$-stacks over the complex-analytic site?

- Discussion Type
- discussion topicinvolution
- Category Latest Changes
- Started by bgm
- Comments 1
- Last comment by bgm
- Last Active Jul 23rd 2020

- Discussion Type
- discussion topicdiscrete and codiscrete topology
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by David_Corfield
- Last Active Jul 23rd 2020

created discrete and codiscrete topology

- Discussion Type
- discussion topicJónsson-Tarski topos
- Category Latest Changes
- Started by jonsterling
- Comments 2
- Last comment by Mike Shulman
- Last Active Jul 23rd 2020

- Discussion Type
- discussion topicAnderson duality
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jul 22nd 2020

started a minimum at

*Anderson duality*just for compleness, see the other thread on*dualizing object in a closed category*.

- Discussion Type
- discussion topiclens space
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Richard Williamson
- Last Active Jul 22nd 2020

- Discussion Type
- discussion topicdifferentiable stack
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 22nd 2020

I have added pdf-links to the reference

- David Carchedi,
*Categorical Properties of Topological and Diffentiable Stacks*, PhD thesis, Universiteit Utrecht, 2011 (dspace:1874/208971, pdf)

and promoted this to the top of the list, since I suppose this is the most comprehensive account that a reader might want to go to first. Will also edit accordingly at

*topological stack*- David Carchedi,

- Discussion Type
- discussion topicHandbook of Algebraic Topology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 22nd 2020

- Discussion Type
- discussion topicJohann Sigurdsson
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 22nd 2020

breif

`category:people`

-entry for hyperlinking references at*parametrized stable homotopy theory*

- Discussion Type
- discussion topicČech nerve
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jul 21st 2020

- Discussion Type
- discussion topicMichael Kapovich
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2020

brief

`category:people`

-entry for hyperlinking references at*hyperbolic manifold*and*orbifold*

- Discussion Type
- discussion topicpushout-product axiom
- Category Latest Changes
- Started by Hurkyl
- Comments 4
- Last comment by Urs
- Last Active Jul 21st 2020

- Discussion Type
- discussion topicdiscrete fibration
- Category Latest Changes
- Started by zskoda
- Comments 7
- Last comment by Tim_Porter
- Last Active Jul 21st 2020

In discrete fibration I added a new section on the Street’s definition of a discrete fibration from $A$ to $B$, that is the version

**for spans of internal categories**. I do not really understand this added definition, so if somebody has comments or further clarifications…

- Discussion Type
- discussion topicequivariant differential K-theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2020

- Discussion Type
- discussion topicorbifold differential K-theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2020

- Discussion Type
- discussion topicKrzysztof Galicki
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 21st 2020

brief

`category:people`

-entry for hyperlinking references at*Sasakian geometry*