you can define $\Cat$ to be the 2-category of all $U'$-small categories, where $U'$ is some Grothendieck universe containing $U$. That way, you have $\Set \in \Cat$ without contradiction.

Do you agree with changing this to

” you can define $\Cat$ to be the 2-category of all $U'$-small categories, where $U'$ is some Grothendieck universe containing $U$. That way, for every small category $J$, you have the category $\Set^J$ an object of $\Cat$ without contradiction. This way, e.g. the diagram in Cat used in this definition of comma categories is defined. “

?

Reason: motivation is to have the pullback-definition of a comma category in (For others, it’s about the diagram here) defined, or rather, having Cat provide a way to make it precise. Currently, the diagrammatic definition can either be read formally, as a device to encode the usual definition of comma categories, or a reader can try to consult Cat in order to make it precise. Then they will first find only the usual definition of Cat having small objects only, which does not take care of the large category

$Set^I$

used in the pullback-definition. Then perhaps they will read all the way up to Grothendieck universes, but find that option not quite sufficient either since it only mentions Set, but not $Set^{Interval}$ . It seems to me that large small-presheaf-categories such as $Set^{Interval}$ can be accomodated, too, though.

(Incidentally, tried to find a “canonical” thread for the article “Cat”, by using the search, but to no avail. Therefore started this one.)

]]>https://ncatlab.org/nlab/show/Récoltes+et+semailles ]]>

I would like to know the great nForum community of "categorical physics" would be interested in contribute. Give it a chance and take a look into the project https://github.com/gcarmonamateo/GeomFormes and hopefully caught your interest in it.

(Sorry for the imprecisions in the English language). ]]>

*C’est au cours de ce travail [“La Longue Marche”] aussi (mais développé dans des notes distinctes) qu’apparaît le thème central de la géométrie algébrique anabélienne, qui est de reconstituer certaines variétés X dites “anabéliennes” sur un corps absolu K à partir de leur groupe fondamental mixte, extension de Gal(K̅/K) par π1(XK̅); c’est alors que se dégage la “conjecture fondamentale de la géométrie algébrique anabélienne”, proche des conjectures de Mordell et de Tate que vient de démontrer Faltings. Esquisse d’un programme*

I am currently working in the transcription (with the collaboration of M. Künzer) of this “Note”. The project is open source and I would like to invite you to contribute.

]]>The article local model structure on simplicial presheaves states that for a site with enough points stalkwise weak equivalences of simplicial presheaves coincide with weak equivalences of simplicial presheaves in the Bousfield localization of componentwise weak equivalences with respect to all hypercovers (i.e., weak equivalences in a local model structure).

Is a proof of this statement written up somewhere? (The article cited above gives a reference to Jardine, which claims, but does not prove this statement.)

Also, is it possible to formulate an analog of this statement for sites that do not have enough points? (Presumably we would have to talk about sufficiently refined (hyper)covers instead of points.)

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