Assume that $\mathsf{C}$ is a small category (in my case with finitely many objects but this is probably irrelevant). In a paper I'm studying at the moment there is a notion used constantly, this of $\pi_1(\mathsf{C})$. This is some kind of (probably!!) fundamental groupoid occurring in that context , but I don't really know what it is and there is no additional information concerning it. Could you please give me a clue (or a reference)?
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1$\begingroup$ Usually it is the groupoid you get by formally inverting each arrow of your category. For instance, if C is a monoid, then $\pi_1(C)$ is the group with generators the elements of $C$ and relations the multiplication of $C$. The universal property of $\pi_1(C)$ is that presheaves on $\pi_1(C)$ correspond to presheaves on $C$ where each arrow is sent to a bijective mapping. $\endgroup$– Benjamin SteinbergCommented Jun 22, 2017 at 17:22
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$\begingroup$ Thank you very much for your comment, this is probably how it's getting used in the paper roughly, so must be this. Is there any specific reason that we use this notation? Is there any chance to be equivalent with the fundamental groupoid as category of some simplicial set associated to $\mathsf{C}$? Any reference is highly appreciated if you know. $\endgroup$– mayer_vietorisCommented Jun 22, 2017 at 17:36
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4$\begingroup$ Yes, the groupoid described by Benjamin Steinberg is equivalent to the fundamental groupoid of the nerve of C (or, if you prefer actual topological spaces, of the geometric realization of that nerve). $\endgroup$– Omar Antolín-CamarenaCommented Jun 22, 2017 at 17:38
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$\begingroup$ You might look at the book of Gabriel and Zisman $\endgroup$– Benjamin SteinbergCommented Jun 22, 2017 at 17:48
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1$\begingroup$ Thank you very much both I think this is exactly what I need. Whoever wants is welcome to merge up the comments and write out an answer. $\endgroup$– mayer_vietorisCommented Jun 22, 2017 at 17:52
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