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- Discussion Type
- discussion topicinductive cover
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topiclocalic stack
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Guest
- Last Active Nov 16th 2022

- Discussion Type
- discussion topiclocally contractible space
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

At locally contractible space David Roberts had written a conjecture relating local contractibility to local triviality of constant n-stacks. I have added to that a converse proposition.

- Discussion Type
- discussion topiclocalic groupoid
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topictopological groupoid
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topiclocalic group
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topicmorphism
- Category Latest Changes
- Started by J-B Vienney
- Comments 1
- Last comment by J-B Vienney
- Last Active Nov 16th 2022

- Discussion Type
- discussion topiccategories and logic - table
- Category Latest Changes
- Started by nLab edit announcer
- Comments 7
- Last comment by J-B Vienney
- Last Active Nov 16th 2022

- Discussion Type
- discussion topicmonoidal functor
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by J-B Vienney
- Last Active Nov 16th 2022

created (finally) lax monoidal functor (redirecting monoidal functor to that) and strong monoidal functor.

Hope I got the relation to 2-functors right. I remember there was some subtlety to be aware of, but I forget which one. I could look it up, but I guess you can easily tell me.

- Discussion Type
- discussion topicgeometric category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

added to the definiton-section at

*geometric category*a pointer to the alternative terms “logical category” and “pre-logos”.

- Discussion Type
- discussion topicgeometric functor
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

added links to regular hyperdoctrine, coherent hyperdoctrine, and geometric hyperdoctrine

Anonymous

- Discussion Type
- discussion topiccoherent category
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

added to

*coherent category*a brief section*Subobjects, slices and internal logic*.

- Discussion Type
- discussion topiccoherent hyperdoctrine
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

created

*coherent hyperdoctrine*(Will now try to bring the entry

*hyperdoctrine*itself into a little bit of shape…)

- Discussion Type
- discussion topicgeometric hyperdoctrine
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topicregular hyperdoctrine
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topicBoolean hyperdoctrine
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

created an entry

*Boolean hyperdoctrine*, just for completeness and so as to link to Todd’s notes.

- Discussion Type
- discussion topicfirst-order hyperdoctrine
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topichyperdoctrine
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topiclocally positive locale
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topicovert space
- Category Latest Changes
- Started by Mike Shulman
- Comments 11
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

At overt space there was a remark that since the definition quantifies over “spaces”, the overtness of a single space might depend on the general meaning chosen for “space”, but that no example was known to the author. I added an example involving synthetic topology, which may not be quite what the author of that remark was thinking of, but which I think is interesting.

- Discussion Type
- discussion topicpositive locale
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topicmetric locale
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topicproper action
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Nov 16th 2022

- Discussion Type
- discussion topicA1-cohesive homotopy type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Nov 16th 2022

- Discussion Type
- discussion topichadron
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 16th 2022

added graphics of the light hadron masses from Fodor-Hoelbling 12

- Discussion Type
- discussion topicpremetric locale
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 15th 2022

- Discussion Type
- discussion topiccontinuum
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by nLab edit announcer
- Last Active Nov 15th 2022

I have added to continuum a paragraph titled In cohesive homotopy type theory.

This is a simple observation and idea that I have been carrying around for a while. Several people are currently thinking about ways to axiomatize the reals in (homotopy) type theory.

With

*cohesive*homotopy type theory there is what looks like an interesting option for an approach different to the other ones: one can ask more generally about line objects $\mathbb{A}^1$ that look like continua.One simple way to axiomatize this would be to say:

$\mathbb{A}$ is a ring object;

it is geometrically contractible, $\mathbf{\Pi} \mathbb{A} \simeq *$.

The last condition reflects the “continuumness”. For instance in the standard model Smooth∞Grpd for smooth homotopy cohesion, this distibuishes $\mathbb{A} = \mathbb{Z}, \mathbb{Q}$ from $\mathbb{A} = \mathbb{R}, \mathbb{C}$.

So while this axiomatization clearly captures one aspect of “continuum” very elegantly, I don’t know yet how far one can carry this in order to actually derive statements that one would want to make, say, about the real numbers.

- Discussion Type
- discussion topicconnected space
- Category Latest Changes
- Started by Todd_Trimble
- Comments 23
- Last comment by nLab edit announcer
- Last Active Nov 15th 2022

I added a bunch of things to connected space: stuff on the path components functor, an example of a countable connected Hausdorff space, and the observation that the quasi-components functor is left adjoint to the discrete space functor $Set \to Top$ (Wikipedia reports that the connected components functor is left adjoint to the discrete space functor, but that’s wrong).

This bit about quasi-components functor had never occurred to me before, although it seems to be true. I’m having difficulty getting much information on this functor. For example, does it preserve finite products? I don’t know, but I doubt it. Does anyone reading this know?

- Discussion Type
- discussion topicspatial type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 15th 2022

- Discussion Type
- discussion topicdevelopment and current state of mathematics
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 15th 2022

- Discussion Type
- discussion topictwo-sided fibration
- Category Latest Changes
- Started by Mike Shulman
- Comments 24
- Last comment by mattecapu
- Last Active Nov 15th 2022

Finally split two-sided fibration off of Grothendieck fibration. Thanks to Emily Riehl for adding the definitions here.

- Discussion Type
- discussion topicreal-cohesive homotopy type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 15th 2022

- Discussion Type
- discussion topichigher homotopy van Kampen theorem
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Dylan Wilson
- Last Active Nov 15th 2022

I added to higher homotopy van Kampen theorem the statement of the theorem by Jacob Lurie.

- Discussion Type
- discussion topicintegral homotopy theory
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Nov 15th 2022

Renamed to ’integral homotopy theory’ as there are now both analogs of Quillen and Sullivan’s approaches. Added references for latter

{#Horel22} Geoffroy Horel,

*Binomial rings and homotopy theory*(arXiv:2211.02349){#Yuan23} Allen Yuan,

*Integral models for spaces via the higher Frobenius*. Journal of the American Mathematical Society, 36(1):107–175, 2023, (arXiv:1910.00999).

- Discussion Type
- discussion topicShin-ya Katsumata
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 15th 2022

- Discussion Type
- discussion topicSebastian Wolf
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 15th 2022

- Discussion Type
- discussion topicLouis Martini
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 15th 2022

- Discussion Type
- discussion topiccategory object in an (infinity,1)-category
- Category Latest Changes
- Started by Mike Shulman
- Comments 11
- Last comment by Urs
- Last Active Nov 15th 2022

It was pointed out to me today that in the very special case of internal (0,1)-category objects in Set, what we are calling a “pre-category” reduces to a preordered set, while adding the “univalence/Rezk-completeness” condition to make it a “category” promotes it to a partially ordered set. I feel like surely I knew that once, but if so, I had forgotten. It provides some extra weight behind this term “precategory”, especially since some category theorists like to say merely “ordered set” to mean “partially ordered set”.

- Discussion Type
- discussion topicstar-autonomous category
- Category Latest Changes
- Started by Mike Shulman
- Comments 25
- Last comment by J-B Vienney
- Last Active Nov 15th 2022

I added to star-autonomous category a mention of “$\ast$-autonomous functors”.

- Discussion Type
- discussion topicMoonshine
- Category Latest Changes
- Started by Tim_Porter
- Comments 8
- Last comment by Urs
- Last Active Nov 15th 2022

Has anyone noted what is happening at Moonshine?

- Discussion Type
- discussion topicBerkovits superstring
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Nov 15th 2022

In reply to an email from John Huerta, I created Berkovits superstring .

- Discussion Type
- discussion topicMiguel A. Vázquez-Mozo
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 15th 2022

- Discussion Type
- discussion topicGreen-Schwarz mechanism
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 15th 2022

I have expanded Green-Schwarz mechanism a fair bit

- Discussion Type
- discussion topicflat modality
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 14th 2022

added type theoretic rules for the flat modality and copied references from sharp modality

Anonymous

- Discussion Type
- discussion topicposet of subobjects
- Category Latest Changes
- Started by Peter Heinig
- Comments 18
- Last comment by nLab edit announcer
- Last Active Nov 14th 2022

poset of subobjects did not yet have a remark on the frequently occurring question whether to define subobjects as morphisms or as isomorphism

*classes*thereof, nor did it yet (seem) to have a thread. Added both. In particular, added the phrase “groupoid of subobjects”. Added two reference where the morphism-versus-class-of-morphisms issue is discussed. Refrained from making any recommendations or comparisions, only writing “alternative definition”.Whenever I see this topic coming up, I am reminded of the “Isomorphic types are equal?!” debate.

Since I do not have much experience in this debate, I shy away from trying to connect isomorphic-types-are-equal with poset of subobjects, but it seems to me that something relevant could be said in this regard in poset of subobjects by someone more experienced.

- Discussion Type
- discussion topicmonad (in computer science)
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Urs
- Last Active Nov 14th 2022

- Discussion Type
- discussion topicrelation between type theory and category theory
- Category Latest Changes
- Started by Urs
- Comments 97
- Last comment by nLab edit announcer
- Last Active Nov 14th 2022

now I have finally the time to come back to this, as announced, and so I am now starting an entry:

*relation between type theory and category theory*.So far there is just some literature collected. I now plan to extract the essence of Seely’s artice into the entry in some technical detail.

- Discussion Type
- discussion topicmaterial versus structural set theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 14th 2022

moving plenty of material from material set theory that I felt deserved its own page on the difference between material and structural set theory.

Anonymous

- Discussion Type
- discussion topicquasigroupoid
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 14th 2022

- Discussion Type
- discussion topicVarmo Vene
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 14th 2022

- Discussion Type
- discussion topicstructural set theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 31
- Last comment by nLab edit announcer
- Last Active Nov 14th 2022

In response to discussions at set theory, created structural set theory with a tentative formal definition of when a set theory is "structural."

- Discussion Type
- discussion topicmaterial set theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 40
- Last comment by Urs
- Last Active Nov 14th 2022

- Discussion Type
- discussion topicautomaton
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Nov 14th 2022

Added this pointer:

An early discussion of automata via string diagrams in the Cartesian monoidal category of finite sets:

- Günter Hotz,
*Eine Algebraisierung des Syntheseproblems von Schaltkreisen*, EIK, Bd. 1, (185-205), Bd, 2, (209-231) 1965 (part I, part II, compressed and merged pdf)

incuding that new compressed and merged pdf which I produced, as a service to the community.

- Günter Hotz,

- Discussion Type
- discussion topicKleene's fixed point theorem
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Nov 14th 2022

a stub entry, for the moment just to make a link work at

*regular language*

- Discussion Type
- discussion topicmembership relation
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 14th 2022

- Discussion Type
- discussion topicfixed point
- Category Latest Changes
- Started by Todd_Trimble
- Comments 16
- Last comment by Guest
- Last Active Nov 13th 2022

Made a start on an article fixed point, which might need to be farmed out to “sub-pages” (as this is a mighty big general topic).

- Discussion Type
- discussion topicallegorical set theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 13th 2022

- Discussion Type
- discussion topiccategorical set theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 13th 2022

starting page on categorical set theories as those set theories which attempt to directly define the category Set of sets and functions; in contrast to the soon-to-be-created article allegorical set theory which are those set theories which attempt to directly define the allegory Rel of sets and relations.

Anonymous

- Discussion Type
- discussion topicinitial algebra of an endofunctor
- Category Latest Changes
- Started by anqurvanillapy
- Comments 5
- Last comment by maxsnew
- Last Active Nov 13th 2022

- Discussion Type
- discussion topicAdámek's fixed point theorem
- Category Latest Changes
- Started by maxsnew
- Comments 1
- Last comment by maxsnew
- Last Active Nov 13th 2022

Create a page for this theorem (mostly copied the text from initial algebra of an endofunctor).