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- Discussion Type
- discussion topicHaldun Özgür Bayındır
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 25th 2023

- Discussion Type
- discussion topicBenson Farb
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 25th 2023

- Discussion Type
- discussion topicThomas Church
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 25th 2023

- Discussion Type
- discussion topicRita Jimenez Rolland
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 25th 2023

- Discussion Type
- discussion topicwalking isomorphism
- Category Latest Changes
- Started by Richard Williamson
- Comments 17
- Last comment by Urs
- Last Active Apr 25th 2023

Little page to focus on this important notion, as opposed to the general remarks at walking structure.

- Discussion Type
- discussion topic(infinity,0)-category
- Category Latest Changes
- Started by jesuslop
- Comments 2
- Last comment by Urs
- Last Active Apr 25th 2023

- Discussion Type
- discussion topiclocally
- Category Latest Changes
- Started by varkor
- Comments 5
- Last comment by Urs
- Last Active Apr 25th 2023

- Discussion Type
- discussion topicinfinity-stack
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Apr 25th 2023

- Discussion Type
- discussion topiclocal adjunction
- Category Latest Changes
- Started by varkor
- Comments 5
- Last comment by Urs
- Last Active Apr 25th 2023

- Discussion Type
- discussion topicparametric right adjoint
- Category Latest Changes
- Started by Mike Shulman
- Comments 20
- Last comment by BryceClarke
- Last Active Apr 24th 2023

Added alternative terminology “local right adjoint” and “strongly cartesian monad” from Berger-Mellies-Weber. They claim the former “has become the more accepted terminology” than “parametric right adjoint”; does anyone know other references to support this? (I think it’s certainly more logical, in that it fits with the general principle of “local” meaning “on slice categories” — not to be confused with the different general principle of “local” meaning “in hom-objects”.)

- Discussion Type
- discussion topicneural network
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by David_Corfield
- Last Active Apr 24th 2023

Stub. For the moment just for providing a place to record this reference:

- Jean Thierry-Mieg,
*Connections between physics, mathematics and deep learning*, Letters in High Energy Physics, vol 2 no 3 (2019) (doi:10.31526/lhep.3.2019.110)

- Jean Thierry-Mieg,

- Discussion Type
- discussion topicstuff, structure, property
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Urs
- Last Active Apr 24th 2023

I have removed the following discussion box from

*stuff, structure, property*– because the entry text above it no longer contained the word that the discussion is about :-)

[begin forwarded discussion]

+–{: .query} Mike: Maybe you all had this out somewhere that I haven’t read, but in the English I am accustomed to speak, “property” is not a mass noun. So you can “forget a property” or “forget properties” but you can’t “forget property.”

*Toby*: Well, ’property’*can*be a mass noun in English, but not in this sense. Also, if we were to invent an entirely new word for the concept, it would surely be a mass noun. Together, these may explain why it's easy to slip into talking this way, but I agree that it's probably better to use the plural count noun here. =–[end forwarded discussion]

- Discussion Type
- discussion topicdimension
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by nLab edit announcer
- Last Active Apr 24th 2023

created stub for dimension

- Discussion Type
- discussion topicenriched groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 24th 2023

- Discussion Type
- discussion topicAdS-QCD correspondence
- Category Latest Changes
- Started by Urs
- Comments 84
- Last comment by Urs
- Last Active Apr 24th 2023

- Discussion Type
- discussion topicAlexis Virelizier
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 24th 2023

- Discussion Type
- discussion topicAlain Bruguières
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 24th 2023

- Discussion Type
- discussion topicCartesian morphism
- Category Latest Changes
- Started by Hurkyl
- Comments 8
- Last comment by Dmitri Pavlov
- Last Active Apr 24th 2023

I felt like the (∞,1) section should give an abstract description rather than a model specific one, so I’ve done so, and proved in the abstract the equivalence between the hom-space and slice-category characterizations.

It feels like cheating to invoke the Grothendieck construction for it; can the argument be made just as cleanly without it?

… I’m having trouble with the formatting, so I’m going to do some bisection to track down the issue….

- Discussion Type
- discussion topictheory of presheaf type
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Thomas Holder
- Last Active Apr 23rd 2023

- Discussion Type
- discussion topiclocalization of model categories
- Category Latest Changes
- Started by Tim_Porter
- Comments 14
- Last comment by Dmitri Pavlov
- Last Active Apr 23rd 2023

I have expanded the Idea section in localization of model categories as it previously seemed to be a stub. (It said: A localisation for model categories. Doh!) I have given a quote from Hirschhorn’s book.

- Discussion Type
- discussion topicnuclear adjunction
- Category Latest Changes
- Started by varkor
- Comments 6
- Last comment by varkor
- Last Active Apr 23rd 2023

- Discussion Type
- discussion topicsuperalgebra
- Category Latest Changes
- Started by Urs
- Comments 42
- Last comment by perezl.alonso
- Last Active Apr 23rd 2023

started working on superalgebra. But have to interrupt now.

- Discussion Type
- discussion topicRudolf Carnap
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Apr 23rd 2023

- Discussion Type
- discussion topicYang-Mills instanton
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by nLab edit announcer
- Last Active Apr 23rd 2023

added to Yang-Mills instanton a discussion of instantons as tunnelings between Chern-Simons vacua.

- Discussion Type
- discussion topicLuca Pol
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 23rd 2023

- Discussion Type
- discussion topicJordan Williamson
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 23rd 2023

- Discussion Type
- discussion topicMichal Przybylek
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 23rd 2023

- Discussion Type
- discussion topicstrict epimorphism
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Apr 23rd 2023

Added:

- Ion Bucur, Aristide Deleanu,
*Introduction to the theory of categories and functors*, Wiley, 1968.

- Ion Bucur, Aristide Deleanu,

- Discussion Type
- discussion topicpasting law for pullbacks
- Category Latest Changes
- Started by Tobias Fritz
- Comments 8
- Last comment by varkor
- Last Active Apr 22nd 2023

- Discussion Type
- discussion topicJean-Pierre Tignol
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 22nd 2023

- Discussion Type
- discussion topiccomplex number
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 22nd 2023

- Discussion Type
- discussion topicGalois theory
- Category Latest Changes
- Started by Urs
- Comments 35
- Last comment by Urs
- Last Active Apr 22nd 2023

the entry Galois theory used to be a stub with only some links. I have now added plenty of details.

- Discussion Type
- discussion topiclax-idempotent 2-monad
- Category Latest Changes
- Started by John Baez
- Comments 4
- Last comment by varkor
- Last Active Apr 22nd 2023

I corrected an apparent typo:

A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta \circ a$

to

A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta_A \circ a$

It might be nice to say $\eta_A$ is the unit of the algebra….

- Discussion Type
- discussion topiclarge cocompleteness
- Category Latest Changes
- Started by varkor
- Comments 6
- Last comment by varkor
- Last Active Apr 22nd 2023

Started this page to record various facts about large cocompletions as I survey some of the literature. It’s possible that some of these concepts should have their own pages eventually, but at the moment this seems awkward as many don’t already have names in the literature (or have names that are not ideal for various reasons).

- Discussion Type
- discussion topicJohn Isbell
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 22nd 2023

- Discussion Type
- discussion topicIsbell duality
- Category Latest Changes
- Started by spitters
- Comments 43
- Last comment by Urs
- Last Active Apr 22nd 2023

http://ncatlab.org/nlab/show/Isbell+duality

Suggests that Stone, Gelfand, … duality are special cases of the adjunction between CoPresheaves and Presheaves. A similar question is raised here. http://mathoverflow.net/questions/84641/theme-of-isbell-duality

However, this paper http://www.emis.ams.org/journals/TAC/volumes/20/15/20-15.pdf

seems to use another definition. Could someone please clarify?

- Discussion Type
- discussion topictotal category
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Apr 22nd 2023

at

*total category*I have added after the definition and after the first remark these two further remarks:

+– {: .num_remark}

###### Remark

Since the Yoneda embedding is a full and faithful functor, a total category $C$ induces an idempotent monad $Y \circ L$ on its category of presheaves, hence a modality. One says that $C$ is a totally distributive category if this modality is itself the right adjoint of an adjoint modality.

=–

+– {: .num_remark}

###### Remark

The $(L \dashv Y)$-adjunction of a total category is closely related to the $(\mathcal{O} \dashv Spec)$-adjunction discussed at

*Isbell duality*and at*function algebras on ∞-stacks*. In that context the $L Y$-modality deserves to be called the*affine modality*.=–

- Discussion Type
- discussion topiccrossed group
- Category Latest Changes
- Started by Thomas Holder
- Comments 2
- Last comment by Thomas Holder
- Last Active Apr 22nd 2023

Created a seperate page with material mostly copied over from generalized Reddy category plus some additional references.

- Discussion Type
- discussion topiclocalization of a simplicial model category > history
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active Apr 22nd 2023

This article describes left Bousfield localizations. Why is it separate from left Bousfield localization? I suggest deleting this article.

- Discussion Type
- discussion topiccategories mailing list
- Category Latest Changes
- Started by varkor
- Comments 3
- Last comment by Urs
- Last Active Apr 22nd 2023

- Discussion Type
- discussion topicTheory and Applications of Categories
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 8
- Last comment by DavidRoberts
- Last Active Apr 22nd 2023

- Discussion Type
- discussion topicprofunctor
- Category Latest Changes
- Started by Mike Shulman
- Comments 31
- Last comment by John Baez
- Last Active Apr 21st 2023

How would people feel about renaming distributor to profunctor? I seem to recall that when this came up on the Cafe, I was the main proponent of the former over the latter, and I've since changed my mind.

- Discussion Type
- discussion topicMalcev variety
- Category Latest Changes
- Started by Todd_Trimble
- Comments 5
- Last comment by Sam Staton
- Last Active Apr 21st 2023

I added some material to Mal’cev variety, namely proofs showing the various characterizations are equivalent, and a brief Examples section.

- Discussion Type
- discussion topiccoreflective subcategory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Apr 21st 2023

all of

`[[!redirects coreflector]] [[!redirects coreflectors]] [[!redirects coreflection]] [[!redirects coreflections]] [[!redirects coreflective subcategory]] [[!redirects coreflective subcategories]]`

used to still be in

*reflective subcategory*. I have removed it there and instead included these redirects here

- Discussion Type
- discussion topicarity class
- Category Latest Changes
- Started by Mike Shulman
- Comments 9
- Last comment by varkor
- Last Active Apr 21st 2023

Created arity class. Added links from a few places, but there are probably others I didn’t think of.

- Discussion Type
- discussion topicGalois representation
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by zskoda
- Last Active Apr 21st 2023

added pointer to

- Tom Lovering,
*Etale cohomology and Galois Representations*, 2012 (pdf)

for review of how Galois representations are arithmetic incarnations of local systems/flat connections. Added the same also to

*local system*and maybe elsewhere.- Tom Lovering,

- Discussion Type
- discussion topicPyotr Sergeyevich Novikov
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 21st 2023

- Discussion Type
- discussion topicLean
- Category Latest Changes
- Started by Urs
- Comments 52
- Last comment by zskoda
- Last Active Apr 21st 2023

added references to

*Lean*

- Discussion Type
- discussion topicAndrés Villaveces
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 21st 2023

- Discussion Type
- discussion topicsymplectic manifold
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by David_Corfield
- Last Active Apr 21st 2023

I have expanded symplectic manifold a little

- Discussion Type
- discussion topicVardan Oganesyan
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 21st 2023

- Discussion Type
- discussion topicgeneralized Reedy category
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Apr 21st 2023

added to

*generalized Reedy category*a bunch of definitions and propositions from Cisinski’s article, concerning the notion of normal morphisms of presheaves over a generalized Reedy category.

- Discussion Type
- discussion topicHeraclitus
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Apr 21st 2023

- Discussion Type
- discussion topicone-point compactification
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Urs
- Last Active Apr 21st 2023

tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.

- Discussion Type
- discussion topicmodality
- Category Latest Changes
- Started by Urs
- Comments 55
- Last comment by nLab edit announcer
- Last Active Apr 20th 2023

added to

*modality*a minimum of pointers to the meaning in philosophy (Kant).

- Discussion Type
- discussion topicexact functor
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by varkor
- Last Active Apr 20th 2023

added to exact functor the characterization of left exact functors as those preserving terminal object and pullbacks. This was previously stated only at finitely complete category.

- Discussion Type
- discussion topicsmall presheaf
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by varkor
- Last Active Apr 20th 2023

Added a reference.

Can we say exactly what kind of pretopos the category of small presheaves on a category C is?

Is it a ΠW-pretopos, provided that PC is complete?

- Discussion Type
- discussion topicconnected object
- Category Latest Changes
- Started by Todd_Trimble
- Comments 6
- Last comment by varkor
- Last Active Apr 20th 2023

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 topicadjoint functor theorem
- Category Latest Changes
- Started by John Baez
- Comments 16
- Last comment by varkor
- Last Active Apr 20th 2023

I fixed a trivial typo in adjoint functor theorem but left wondering about this:

… the limit

$L c := \lim_{c\to R d} d$over the comma category $c/R$ (whose objects are pairs $(d,f:c\to R d)$ and whose morphisms are arrows $d\to d'$ in $D$ making the obvious triangle commute in $C$) of the projection functor

$L c = \lim_{\leftarrow} (c/R \to D ) \,.$I don’t really understand this (and while I could figure it out, it’s probably not good to make readers do so). At first it sounds like someone is saying “the limit $L c$ over the comma category of the projection functor $L c$”, which would be circular. But it must be that both formulas are intended as synonymous definitions of $L c$. At that point one is left wondering why one has a backwards arrow under it and the other does not. I guess old-fashioned people prefer writing limits with backwards arrows under them, so someone is trying to cater to all tastes? I think it’s better in this website to use $lim$ and $colim$ for limit and colimit.

I could probably guess how to fix this, but I won’t since I might screw something up.

- Discussion Type
- discussion topicMaru Sarazola
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 20th 2023