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Given a total function $f \colon A \to B$, by grouping the "points" on $A$ which have the same image on $B$, namely $a,b \in A$ located in the same equivalence class $G_{i}$ iff $f(a) = f(b)$.

By that way, I get a partition $G = \{ G_{1},G_{2},\dotsc \}$ on $A$ derived from $f$, and concurrently I can represent $f = g \circ h$ where $h \colon A \to G$ and $g \colon G \to B$. This representation is unique if $h$ is monic, namely if $f = g' \circ h'$ where $g' \colon A \to G$ and monic $h' \colon G \to B$, then $g' = g$ and $h' = h$.

I doubt that this simple situation has a more general interpretation in the category theory but I don't know what it is. Can anyone please give me some suggestion ?

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    $\begingroup$ This equivalence relation is called the kernel pair of $f$. $\endgroup$
    – Zhen Lin
    Commented May 30, 2013 at 11:35

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Actually $G$ is the image of $f$. This can be stated categorically, see the entry in the nLab, but it depends on the definition of “subobjects”.

If you are interested in the relation between $f$ and the equivalence relation you mentioned, have a look at universal algebra and its general version of the isomorphism theorem—but that is not a categorical topic.

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  • $\begingroup$ Thank you very much for the detail response. I admit that I spent a lot of time to digest the article at nLab because it is not obvious for me at all. $\endgroup$
    – Ta Thanh Dinh
    Commented Jun 3, 2013 at 12:14
  • $\begingroup$ I beg to differ to a point in The User's reply. The general version of the isomorphism theorem IS most definitely a categorical topic. That it can also be handle (in specific (and quite general) cases with no mention of category theory does not mean it is not a categorical topic. Of course, it is also a topic in algebra, and so on. $\endgroup$
    – Tim Porter
    Commented Aug 31, 2014 at 9:52
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Any morphism, $f:A\to B$, in a nice enough category (e.g. with finite limits) will define a pullback whose two projections to $A$ give many of the features of an equivalence relation. This is of central importance to many parts of category theory. Look at ideas such as effective epimorphism, and its variants in the n-Lab, or a standard (> medium depth) source for category theory.

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  • $\begingroup$ I appreciate your answer though I am quite not sure I fully understand it. It is even more abstract than the response above. $\endgroup$
    – Ta Thanh Dinh
    Commented Jun 3, 2013 at 12:22
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    $\begingroup$ The idea is that if $f:A\to B$, then you form the subset of the product $E(f) = \{(a_1,a_2)\mid f(a_1)=f(a_2)\}\subset A\times A$. This comes with two projections to $A$, inherited from those of the product. This looks a bit like an equivalence relation (and is one e.g. in the case if you are working in the category of Sets), ... but you can do this in any category in which pullbacks exist. $\endgroup$
    – Tim Porter
    Commented Jun 3, 2013 at 13:19

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