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.)

]]>Maybe one day it would be fun to find the time to create as $n$Lab entries all those linked to here:

The WIki History of the Universe,

maybe at least up to the entry “star formation”, which still fits under the “Physics” headline of the $n$Lab (after that geology and then biology kicks in, which we should probably leave to other wikis).

One interesting thought to explore might be if we can expand that link list to the left. ;-)

]]>I think we should have a page generally about “change of universe” for categories, i.e. starting from a(n n-)category that is large (and often bicomplete, monoidal, etc.) w.r.t. one universe and producing a corresponding one that is large (bicomplete, monoidal) w.r.t. a bigger universe. What we currently call the huge (infinity,1)-sheaf (infinity,1)-topos is a special case of this. I’m not sure what to call such a page, though — change of universe? universe enlargement?

(As a side note, I personally prefer “very large” over “huge,” since “huge” is an adjective that has a separate meaning when applied to cardinal numbers, but that’s not a big deal. Perhaps it would be more future-compatible to say “2-large” with reference to the second universe, anticipating a potential need for 3-large etc. categories, but that might have the undesired implication of some relationship with 2- and 3-categories, so maybe it’s a bad idea.)

It would also be nice to have a standard notation for this operation. I (and many other category theorists, I believe) tend to use levels of capitalization of category names, e.g. Set is the category of small sets and SET the category of large sets. Sometimes people write “set” for the category of small sets, “Set” for the category of large ones, and then maybe SET can be used for the category of very large ones. But obviously this doesn’t work well for a generic category name like $\mathcal{C}$. In Kelly’s “basic concepts” he writes $V'$ for the universe-enlargement of the category $V$, while Lurie seems to put hats on things, but both of those notation have lots of other overloaded meanings. One possible notation that just occurred to me is $C^\uparrow$, which looks a bit like a hat and conveys the sense of “raising up” or “making bigger” while not AFAIK clashing with anything else. Thoughts?

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