Editing Groupoid (section)

=== Groupoids in Grpd ===
{{Main|Double groupoid}}
There is an additional structure which can be derived from groupoids internal to the category of groupoids, '''double-groupoids'''.<ref>{{cite arXiv|last1=Cegarra|first1=Antonio M.|last2=Heredia|first2=Benjamín A.|last3=Remedios|first3=Josué|date=2010-03-19|title=Double groupoids and homotopy 2-types|class=math.AT|eprint=1003.3820}}</ref><ref>{{Cite journal|last=Ehresmann|first=Charles|date=1964|title=Catégories et structures : extraits|url=http://www.numdam.org/item/?id=SE_1964__6__A8_0|journal=Séminaire Ehresmann. Topologie et géométrie différentielle|language=en|volume=6|pages=1–31}}</ref> Because '''Grpd''' is a 2-category, these objects form a 2-category instead of a 1-category since there is extra structure. Essentially, these are groupoids <math>\mathcal{G}_1,\mathcal{G}_0</math> with functors<blockquote><math>s,t: \mathcal{G}_1 \to \mathcal{G}_0</math></blockquote>and an embedding given by an identity functor<blockquote><math>i:\mathcal{G}_0 \to\mathcal{G}_1</math></blockquote>One way to think about these 2-groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally. For example, given squares<blockquote><math>\begin{matrix}
\bullet & \to & \bullet \\
\downarrow & & \downarrow \\
\bullet & \xrightarrow{a} & \bullet
\end{matrix}
</math> and <math>\begin{matrix}
\bullet & \xrightarrow{a} & \bullet \\
\downarrow & & \downarrow \\
\bullet & \to & \bullet
\end{matrix}</math></blockquote>with <math>a</math> the same morphism, they can be vertically conjoined giving a diagram<blockquote><math>\begin{matrix}
\bullet & \to & \bullet \\
\downarrow & & \downarrow \\
\bullet & \xrightarrow{a} & \bullet \\
\downarrow & & \downarrow \\
\bullet & \to & \bullet
\end{matrix}</math></blockquote>which can be converted into another square by composing the vertical arrows. There is a similar composition law for horizontal attachments of squares.