# How to understand the graph in the category?

I am reading the Mac Lane"s book: Categories for the Working Mathematician now,but I do not understand the "graph" in it.What is the different from the graph in the category and from the module theory?How does the graph in the category generate the free category?Please give me some suggestions.

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Can you please give a reference of the precise page where you read this? Actually, there are a lot of matches of the word "graph" into CWM... – Fosco Loregian May 2 '11 at 8:48
Just in Ch3.And the graph from the module theory means something like 5-lemma,3by3 lemma,maybe I should say the graph in the homological algebra. – Strongart May 4 '11 at 6:10

I don't know the graph in module theory, so I cannot answer your question there. A category has a lot of structure, namely

It has objects it has arrows it has composition. It has identities

A graph is a category where you don't take into account the last two things on this list. So you just have a list of objects and some arrows between them.

You can formally add compositions of these arrows, and add the identities. You will get a category in this way. This is the free category.

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Thank you,maybe you can show me some nonfree examples to compare with the free category which is generated by the graph. – Strongart May 7 '11 at 5:52
Take a category with two objects $A$ and $B$. Suppose there are two non-identity morphisms $f:A \rightarrow B$ and $g:B\rightarrow A$. This means that $fg=id_B$ and $gf=id_A$. This category is non free. If you start with a graph with two arrows as above, the free category will have an infinite number of morphisms (formally things of the form $fgfgfgfgfgf$). – Thomas Rot May 8 '11 at 20:17
I see,Thank you. – Strongart May 16 '11 at 10:24

A graph as Mac Lane uses it, is also called a 'multidigraph (with loops allowed)', or 'quiver' in graph theory. Every category has (by definition) such an underlying graph: the objects are the vertices, the arrows (morphisms) are the directed arrows, and for every vertex there is a loop, namely the identity morphism.

I believe the term quiver is mainly used in representation theory, and this is probably the thing you mean by the term 'graph' in module theory.

Also see the graph page at the nLab.

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Thank you,but I do not know much about the representation theory,maybe you can show me some nonfree examples to compare with the free category which is generated by the graph. – Strongart May 5 '11 at 10:58