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Given a free group $F_X$, and it's Cayley graph, $Cay(F_X,X)$, we can take the quotient graph of $Cay(F_X,X)$ by the action of $F_X$ on it and obtain $R_{|X|}$, a "bouquet of circles" or "rose" consisting of a single vertex with $|X|$ edges labeled by the generators of $F_X$.

When learning this, I was stricken by the similarity of the bouquet of circles with the categorical definition of a group: Namely, a group may be considered a category with a single object in which all morphisms are isomorphisms. If we draw the object of the category as a vertex and the morphisms as directed edges, we obtain a similar graph (although, clearly, there is a difference, since in the case of the category-based graph, edges are labeled by elements, while in the case of the rose, edges are labeled by generators).

Is there any relationship between these two constructions, or is this merely a coincidence?

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up vote 2 down vote accepted

Sure. The construction you want is called the free category on a directed graph. This is the category freely generated by composing all of the arrows in the directed graph, and running this construction on $|X|$ arrows attached to a single object you get (the category corresponding to) the free monoid on $X$. Throwing in inverses gets you the free group, but this is subtle in general.

Alternately, you can take the geometric realization of your bouquet (that is, realize it as a topological space obtained by wedging together circles) and then take the fundamental group of the result.

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