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The thread Category theory vs order theory quickly really became Topological spaces vs locales, so I’m putting this in a new thread.
At category theory vs order theory, I had originally put in the analogy with category : poset :: strict category : proset. Mike changed this to to category : proset :: skeletal category : poset. I disagree. A proset has two notions of equivalence: the equality of the underlying set, and the symmetrisation of the order relation; a poset has only one. Similarly, a strict category has two notions of equivalence: the equality of the set of objects, and the isomorphism relation; a category has only one. I’m OK with using skeletal categories to compare with posets, since this will make sense to people who only know the evil notion of strict category, but I insist on using strict categories to compare with prosets. So now its strict category : proset :: skeletal category : poset.
Okay, I guess I see what you were getting at. But I think any analogy which includes non-strict categories is going to be misleading as long as we don’t have a non-strict order-theoretic notion to compare them with. I usually think of “proset” as synonymous with thin category, but if you want to insist that a proset always has an underlying set and hence is a strict thin category, I can live with that. But maybe then the pages proset and thin category should be clarified.
(By the way, the image at thin category is giant and fills my entire screen. Is that image really even necessary? I feel like “any two parallel morphisms are equal” is a simple enough concept not to need a diagram.)
I think that thin category is already clear enough, but I added another remark.
Yes, the picture there is huge! As I recall, I had trouble doing it in Codecogs (I tried again and it still isn’t working), so I made it another way, it came out that size, and I was tired of fiddling with it.
I think that the latter half of the Idea section of preorder was terribly misleading, so I rewrote it (and added other clarifying remarks).
The non-strict order-theoretic notion is simply the notion of a poset. Of course, if you start with sets as primary, you’ll always define a poset as a set (complete with equality relation) first and then add in the extra structure of the order relation, which then must be compatible with the equality relation. From a really weak perspective, you would put in the structure of the order relation first (requiring only reflexivity and transitivity) and define the equality relation from that. However, the final results are indistinguishable. Still, this may explain why people like Matthieu Dupont prefer to use the less loaded term ‘order’ instead. (It is allowed to refer to the isomorphism relation in a thin category —that is a poset— as equality because the -groupoid of equivalences between two given objects is in fact a -groupoid. Even for a thin strict category —that is a proset— one could do this, speaking of tight and loose equality, but this might test the sanity of readers who are only used to set-theoretic foundations and think of equality, at least within a given type, as an absolute notion.)
Certainly, non-strict thin categories, up to equivalence, are the same as posets, up to isomorphism. But I think it is very important not to alienate readers whose only foundational experience is with set theory and for whom “is” means, by default, identity rather than equivalence. I started to expand and reorder your comments with this goal in mind, but it started getting much too involved for an “Idea” section, so I moved it to a later section “Relation to thin categories”.
As for the image, how about this?
Image fixed at thin category.
Your edits to preorder seem very clear.
Great, I’m glad we reached an agreement!
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