Editing Groupoid (section)

=== Category-theoretic ===
A groupoid is a [[category (mathematics)#Small and large categories|small category]] in which every [[morphism]] is an [[isomorphism]], i.e., invertible.<ref name="dicks-ventura-96"/> More explicitly, a groupoid <math>G</math> is a set <math>G_0</math> of ''objects'' with
* for each pair of objects ''x'' and ''y'', a (possibly empty) set ''G''(''x'',''y'') of ''morphisms'' (or ''arrows'') from ''x'' to ''y''; we write ''f'' : ''x'' → ''y'' to indicate that ''f'' is an element of ''G''(''x'',''y'');
* for every object ''x'', a designated element <math>\mathrm{id}_x</math> of ''G''(''x'', ''x'');
* for each triple of objects ''x'', ''y'', and ''z'', a [[function (mathematics)|function]] {{tmath|1= \mathrm{comp}_{x,y,z} : G(y, z)\times G(x, y) \rightarrow G(x, z): (g, f) \mapsto gf }};
* for each pair of objects ''x'', ''y'', a function <math>\mathrm{inv}: G(x, y) \rightarrow G(y, x): f \mapsto f^{-1}</math> satisfying, for any ''f'' : ''x'' → ''y'', ''g'' : ''y'' → ''z'', and ''h'' : ''z'' → ''w'':
** {{tmath|1= f\ \mathrm{id}_x = f }} and {{tmath|1= \mathrm{id}_y\ f = f }};
** {{tmath|1= (h g) f = h (g f) }};
** <math>f f^{-1} = \mathrm{id}_y</math> and {{tmath|1= f^{-1} f = \mathrm{id}_x }}.

If ''f'' is an element of ''G''(''x'',''y''), then ''x'' is called the '''source''' of ''f'', written ''s''(''f''), and ''y'' is called the '''target''' of ''f'', written ''t''(''f'').

A groupoid ''G'' is sometimes denoted as {{tmath|1= G_1 \rightrightarrows G_0 }}, where <math>G_1</math> is the set of all morphisms, and the two arrows <math>G_1 \to G_0</math> represent the source and the target.

More generally, one can consider a [[groupoid object]] in an arbitrary category admitting finite fiber products.