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Denote $Pf$ the category of sets and partial functions and $Set_*$ the category of pointed sets.

i can't see why, i was trying some contradiction argument with the definition of isomorphism, maybe taking some particular sets.... supose exists functors $F: Pf \rightarrow Set_*$ and $G: Set_* \rightarrow Pf$ such that $F \circ G = I$ and $G \circ F = I$, i was thinking in find some obligatory definitions, but appear to me that don't exists.

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  • $\begingroup$ I wouldn't try contradiction. I'd try to find a property which is not shared by these categories. $\endgroup$
    – Abellan
    Feb 25 '15 at 21:35
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The two categories in question are equivalent, so if they are not isomorphic, they must differ in an "evil" way.

Hint. Both categories have zero objects. How many zero objects are there in the category of pointed sets? How many zero objects are there in the category of sets and partial functions?

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  • $\begingroup$ ow i see, the empty set is a zero object in the case we consider partial functions.... finite x infinity right? $\endgroup$
    – Oliver
    Feb 25 '15 at 22:10
  • $\begingroup$ I would replace the word "evil" by "set-theoretic" or "0-categorical". $\endgroup$ Apr 7 '15 at 21:38
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Maybe I'm wrong but I'll give it a try:

In $Set_*$ every singleton (a,a) is a zero object. If both categories are isomorphic , the number of zero objects must be the same. The functor $F$ then must send every zero object to a zero object.

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  • $\begingroup$ This two categories are equivalent... this was't supose to happen if we have a equivalence? $\endgroup$
    – Oliver
    Feb 25 '15 at 21:59

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