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moving the following old discussion from truth value to here
+–{.query} Urs: Can’t this be motivated more systematically?
Mike: There’s a funny weirdness that happens at the bottom with the relationship between sets and posets. I don’t completely understand it, nor do I know any reason why “we should expect” the $0$-category of $(-1)$-groupoids to be a $(0,1)$-category, since the 1-category of 0-groupoids (sets) is not a $(1,2)$-category (except trivially). On the other hand, the 1-category of $(0,1)$-categories is a $(1,2)$-category, so maybe the point is that every $(-1)$-groupoid is actually a $(-1,0)$-category?
I don’t really understand why we have separate pages for $(-1)$-groupoids and $(-1)$-categories; has anyone ever come up with a way in which they are different?
Toby: There are separate pages truth value, (-1)-groupoid, (-1)-category and (eventually) 0-poset. Partly it's that (-1)-category was written first, although it's the worst name. But I think that it will be good, given the nature of this wiki, to have pages organised more by term than subject (I will write a bit about this on the Café). But this means that the pages will have different purposes.
As truth value is the most mundane and ordinary name for the concept, that page will have the most content. There we should discuss what a truth value is in a topos or other sort of category, or what it is to a constructivist or other sort of alternative mathematician; there we should discuss how truth values form a poset, and what structures and properties this poset has; there we should talk about what categories and groupoids enriched over this poset (with either of the two obvious monoidal structures) are.
At (-1)-category, which exists only because we like $n$-categories so much and so wonder what an $(-1)$-category is, we explain why the concept is a little fishy but the best way to define it is as a truth value. And there we show how this fits the patterns of the periodic table to the extent that it does, and also poitn out those cases where it doesn't.
At (-1)-groupoid, which again exists because that's our special thing, we explain why a $(-1)$-groupoid is a truth value, and how that fits the patterns of the periodic table.
At 0-poset, we'll do exactly the same thing, only thinking about $n$-posets rather than $n$-groupoids.
(If this were Wikipedia, we would not organise things this way. I'll discuss that at the Café.)
In particular, I think that general facts about internalising the concept to categories other than $\Set$ work best at truth value, while they would really fit in here only if we wanted to explain how that reproduces (or, conceivably, turns out to be different from) a general method for internalising the concept of $n$-groupoid. (Probably we will eventually want to talk about that, but as far as I know there's nothing to say about it now.)
By the way, here is the answer to Mike's question about what “we should expect”: the $\infty$-category of $(n,r)$-categories is an $(n+1,r+1)$-category (which I should discuss on (n,r)-category but have not); accordinly, the $\infty$-category of $n$-groupoids is an $(n,1)$-category. In particular, the $\infty$-category of $(-1)$-groupoids is a $(0,1)$-category. =–
Have the punchlines from that discussion been extracted and recorded at the entry? I don’t have time to study this now, but at a glance the discussion looks interesting and deserving of further attention.
What should be made into its own page? I guess you mean, not the adjective ’boolean’, but “Boolean truth value”?
’Boolean’ is used a noun in programming to mean something of type Bool. I guess this is what Richard means.
Yes, exactly. Programming languages often have a type called ’bool’ or ’boolean’, and one says things like a method ’returns a boolean’. I think we could have a page called ’boolean type’ or something.
Oh, oh, yes of course. Thanks.
added missing cross-link with boolean domain
But maybe these entries need to be merged. Currently lattice of truth values and Boolean and variants are redirecting to truth value, while strictly speaking they refer to the type of truth values, hence to boolean domain.
Right, the text in this entry definitely wants it to be about the truth values as such (-1-truncated types), not about the type they form. So hereby I am turning at least the following former redirects
[[!redirects boolean]]
[[!redirects booleans]]
[[!redirects Boolean]]
[[!redirects Booleans]]
to point instead to boolean domain (as suggested already in comment #3 above)
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