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In ncatlab entry for a thin category* reads: "Up to isomorphism, a thin category is the same thing as a proset. Up to equivalence, a thin category is the same thing as a poset"

I would very much appreciate if one could explain me the above.

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Is the issue that you aren't sure what "up to isomorphism" and "up to equivalence" mean or that you aren't sure what "proset" and "poset" mean? –  Qiaochu Yuan May 2 '13 at 21:25
    
I know the deference of isomorphic of categories and equivalent categories {in equivalence we don't necessarily get back to the very same object but an object isomorphic to that}. But why this terminology is used here. Why thin category differs with a partially ordered class? –  user34942 May 2 '13 at 21:37
    
Do you understand the difference between a preorder and a poset (up to isomorphism)? –  Qiaochu Yuan May 2 '13 at 21:49
    
A poset is a preorderd set which satisfy the anti-symmetry axiom; x<y and y<x => x=y –  user34942 May 2 '13 at 22:15
    
proset = preorder? –  Hurkyl May 2 '13 at 23:12

2 Answers 2

up vote 3 down vote accepted

A category $\Bbb A$ is thin if it has at most one arrow $a\to b$ for any pair $(a,b)$ of objects. Define $a\le b$ iff $\exists\, a\to b$ in $\Bbb A$.

We can have different objects $a,b$ such that $a\le b\le a$. So, this definition doesn't necessarily gives a poset, only a preorder. In this case in $\Bbb A$ we have $a\cong b$.

However, if we consider the skeleton of $\Bbb A$, i.e. we keep exactly one object in each isomorphism class, then we get an equivalent category, and the preorder becomes a poset now.

One can show that posets correspond exactly to the skeletal thin categories. Then combining this to the fact that categories are equivalent iff their skeletons are isomorphic, we're ready.

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Well...in my opinion the statement in ncatlab that you quoted is very informal, to say the least. In fact it runs against the (categorial) definition of proset given by the same ncatlab which says:

In set-theoretic foundations, a preordered set is the same as a thin category (a category in which any two parallel morphisms are equal), and it is partially ordered just when it is skeletal.

Indeed, that's all there is to it:

1-a thin category is called a preorder

2- a preorder (that is: a thin category) with no-nontrivial isomorphisms is called a partial order

To be even more explicit:

2 bis - a preorder (that is: a thin category) whose only isomorphisms are the identities, is called a partial order

or, if you prefer

2 ter - a preorder (that is: a thin category) with no isomorphisms between distinct objects, is called a partial order

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