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Added link to (0,1)-category.
Possibly the two pages should just be merged. I think the very concrete definition here (with diagram) would be a nice complement to the more abstract discussion on the other page, and vice versa.
The proset article says that thin categories are prosets. The thin category article says that (0,1)-categories are thin categories. The (0,1)-category article says that posets are (0,1)-categories. But prosets are not the same as posets, so one of these articles is wrong.
I meant to say that:
“The proset article says that thin categories are equivalent to prosets. The thin category article says that (0,1)-categories equivalent to thin categories. The (0,1)-category article says that posets equivalent to (0,1)-categories.”
Re. #3 and #4. Preorders are equivalent (as categories) to posets, so there is no contradiction.
Yes, but the problem is that some of these entries explicitly speak of strict categories defined up to isomorphism, while others don’t dwell on this point.
I just went and adjusted the wording at (0,1)-category.
But if anyone has the energy, it would be good to write a paragraph (maybe even a dedicated entry) which sorts this out very clearly. It should say very clearly that, in category theory lingo, posets are skeletal prosets, and that this distinction matters up to iso of strict categories but not up to equivalence of categories.
also in intensional type theory unless one has something like the univalence axiom, then in general the function cannot be proven to be an equivalence in a preorder, even though is by definition an equivalence in a poset, so posets and preorders are not equivalent in all foundations. And if the type theory doesn’t have quotient sets/quotient types, then one cannot construct the quotient poset from a preorder either.
Okay, so i have started a new entry:
with some details on what’s going on.
I will now link to this from all relevant entries, so that this is the place where we should offer all the explanations and warning that readers have a right to see.
So if you have further subtleties/details/variations to add (such as in #7) please be invited to open this entry and expand.
In the Properties-section (here) I have removed this paragraph:
So mostly we just talk about posets here, but some references want to distinguish these from thin categories. (It is really a question of whether you're working with strict categories, which are classified up to isomorphism, or categories as such, which are classified up to equivalence.)
and replaced it by:
For more on this see at relation between preorders and (0,1)-categories.
[ edit: fixed now, according to discussion below ]
Neat, that new entry looks quite helpful.
In that last link mentioned above, was the reference to relation between type theory and category theory intentional, or did you mean to link to the new entry relation between preorders and (0,1)-categories? The former page is quite large and doesn’t appear to discuss this question.
Oh, sorry, I have fixed it. Yes, of course I meant to link to relation between preorders and (0,1)-categories.
(This comes from me finding nLab pages by starting to type their URL and then Firefox offering autocompletion, and then me not paying attention to what it’s actually offering.)
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