Timeline for What does the $\pi_1(\mathsf{C})$ really mean?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 22, 2017 at 17:52 | comment | added | mayer_vietoris | 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. | |
| Jun 22, 2017 at 17:48 | comment | added | Benjamin Steinberg | You might look at the book of Gabriel and Zisman | |
| Jun 22, 2017 at 17:38 | comment | added | Omar Antolín-Camarena | 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). | |
| Jun 22, 2017 at 17:36 | comment | added | mayer_vietoris | 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. | |
| Jun 22, 2017 at 17:22 | comment | added | Benjamin Steinberg | 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. | |
| Jun 22, 2017 at 17:18 | history | asked | mayer_vietoris | CC BY-SA 3.0 |