Even better would be a video hosting site such as YouTube, Vimeo, Daily Motion etc, for longevity, with an open access license to allow for virtuous copying.

]]>Ah, there it is, thanks! The beginning of that section misled me about its contents.

]]>If that's where John Baez put them to make them available after UCR stopped hosting them, then I think that that's where we should link them. [ETA: Although since this is Jim Dolan's personal web, then it's he who would link to them.]

]]>The second definition of semisimple category on this page is as a “monoidal linear category” with certain properties. But none of the properties or any of the later discussion on this page uses the monoidal structure, nor is it very consistent with the first defintion (as a semisimple abelian category, which makes no reference to monoidality).

]]>What about section 4.8 of Higher Algebra?

]]>I expanded the definition of the tensor product with an explanation of why accessibility is preserved on the right adjoints, which I think is something that Lurie glosses over a little too quickly.

By the way, the page Noncommutative Algebra claims that that paper is “subsumed as a part of monograph Higher Algebra”, but I can’t find this construction of the tensor product of locally presentable ∞-categories anywhere in Higher Algebra. Would it be better to say “mostly subsumed”?

]]>Created sound doctrine as a stub to record relevant references.

]]>coreflective subcategory. I added a few more links, and a note on the counit.

]]>Corrected the relationship of Isabelle, HOL, and HOL Light. Next week I am going to host a course on Isabelle by Makarius (Marius) Wenzel. Perhaps I will find time to add additional content.

]]>Much better! Thanks.

]]>Well, I have a lot of trouble reading the table as-is anyway. I don’t know why, but the top header row doesn’t have column-separator lines, which means that the third and fourth cells run together and for a long time I was reading it as “inclusion of left generated under exact colimits from localizations small objects”, which is just close enough to meaningful to give me a headache trying to figure it out. (-:

Also, I think the headings “generated under colimits from small objects” and “generated under filtered colimits from small objects” are misleading, because the second sounds like a *stronger* condition than the first (since a filtered colimit is, in particular, a limit) whereas in fact being accessible is weaker than being locally presentable. Moreover, the generation in a locally presentable category *is* under filtered colimits.

I tried rearranging the table some by moving large blocks of text out of it (which I think don’t look good in tables) into the header above, adding a column for the locally finitely presentable case.

]]>Possibly being in any category makes a page be non-orphaned.

It doesn't; in fact, if you look at the category list, you can see that everything listed there is listed twice: once for just being in the category, and once for being an orphan in the category.

What’s the problem with deleting a file that deserves to be deleted?

I would prefer to never make that determination. Blanking a page, orphaning it, writing over it … these are all ways to demote the content of a page, declaring it worthless. But they're also all reversible in case of error. Even though in most (maybe all) of these cases, this conclusion is beyond doubt, I'm pleased that we've never done anything on the nLab that removes information about the history of edits to the wiki (beyond what is lost due to inherent features of Instiki). And as long as spam etc is only a tiny portion of that history, we'll never need to.

]]>I've moved the page to true proposition, keeping redirects from truth and similar terms until somebody writes the article outlined by Todd #9.

]]>Mike, yes, that also makes sense, especially when $x$ is not a natural number.

Of course, there's also the case when $k$ is negative; then it's a *rising* power and a shifted *reciprocal* factorial. (In particular, $1/k! = 0^{\underline{-k}}$.)

Hi Toby,

thanks for your reply. I might still not know what your motivation is. Here would be my comments:

I like to reuse them to get them out of category: empty.

But why put them in that category in the first place? Possibly being in any category makes a page be non-orphaned.

Then nothing ever needs to be deleted.

What’s the problem with deleting a file that deserves to be deleted?

But it should all be worthless.

So then this is motivation to not re-use these pages, since all that accomplishes is to make this worthless content be saved indefinitely, in the page history. And not just saved, but saved in a potentially confusing way, since any serious page reusing this will now retain this random trash in its history.

For these reasons I still think what we should really do is make sure that all pages that are deemed worthless (or worse) receive no links and in particular no “category”-label.

]]>Thanks for the alert. I would be happy to add a column saying “generated under finite colimits from finite objects” and then to move the entry “algebraic lattices” to that column. If the entry gets too wide, I think the software automatically produces a scrollbar environment.

But how to organize it? If we insert that new column to the left of the existing “generated under colimits” then the compatibility with the “inclusion of left exact localizations breaks.

Any suggestion for how to handle this? Maybe we should make a second copy of the whole table, now for finite generation.

]]>I added to Pr(infinity,1)Cat the definition of the tensor product, and some remarks about colimits in $Pr(\infty,1)Cat$ that coincide with limits, essentially because it is an $(\infty,2)$-category with local colimits.

]]>Another problem with this table as currently written (which for some reason no longer renders in comment #1, but looks fine at the nLab page locally presentable categories - table) is that algebraic lattices correspond to locally *finitely* presentable categories, not arbitrary locally presentable categories. I would argue that the posetal analogue of an arbitrary locally presentable category is simply a suplattice. (In particular, not every frame is an algebraic lattice, whereas every topos is a locally presentable category.) If we want to include algebraic lattices in the table, then there should be a column for the locally finitely presentable case of the other rows (but that might make the table too wide).

So what is the appropriate CSS file? (I tried all I could find) Or perhaps I got the font name wrong?

]]>In #6 I meant to write $\invamp$, not $\otimes$. I would argue that $\oplus$ deserves the symbol $\vee$ more than $\invamp$ does, because it behaves more like the “or” of classical logic. But sure, I suppose someone might do things differently.

]]>I would have called it a *shifted* factorial, since the difference between it and $k!$ is that it starts at $x$ rather than at $k$.

Not so unreasonable to model a noncommutative ’or’ if you consider how we use it to spell out consequences, e.g., “Do this, or (else) that will happen”. That’s why it appears in relevant logic.

]]>