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I know almost nothing about category theory (I have just skimmed the first chapters of Aluffi's algebra book), reading this question got me thinking... why should someone mostly interested in combinatorics/graph theory learn category theory?

What I am asking for is examples of how knowledge of category theory might be beneficial for someone doing combinatorics.

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You might be interested in combinatorial species, which are most naturally described as an endofunctor on the category of finite sets and bijections. – Chris Taylor Apr 5 '13 at 11:04
Not beneficial. Forget it. – Berci Apr 5 '13 at 11:07
@Berci: could you elaborate on your rather cryptic comment? – Rasmus Apr 5 '13 at 11:18
Refer to Chris Taylor's comment on combinatorial species. – Zhen Lin Apr 5 '13 at 14:54
I've posted a "follow-up" question about combinatorial species here – Carolus Apr 6 '13 at 4:44

If you are interested in becoming a pioneer in a new area of mathematics that involves combinatorics then there is Combinatorial category theory.

László Lovász talks about this in a video interview starting at 1m59s in

Also in his book Large networks and graph limits, chapter 23,, there is a section on categories in which he says: "In graph theory, the use of categories (as a language and also as guide for asking question in a certain way) has been practiced mainly by the Prague school, and has lead to many valuable results; see e.g. the book by Hell and Nešetřil [2004]." (Graphs and Homomorphisms)

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Interesting. I will look into this, definitely! – Carolus Apr 6 '13 at 6:25
Apparently an update for the second link, which is dead: – darij grinberg May 3 '15 at 5:10

You might find interesting Extensive categories and the size of an orbit by Ernie Manes (TAC, 2015).

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