groupoids - MathStructures

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=====Groupoids=====

Abbreviation: **Grpd**

====Definition====
A \emph{groupoid} is a [[category]] $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$ such that

every morphism is an isomorphism: $\forall x\exists y\ x\circ y=\text{dom}(x)\text{ and }y\circ x=\text{cod}(x)$ 

==Morphisms==
Let $\mathbf{C}$ and $\mathbf{D}$ be Schroeder categories. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a \emph{functor}: $h(x\circ y)=h(x)\circ h(y)$, $h(\text{dom}(x))=\text{dom}(h(x))$ and $h(\text{cod}(x))=\text{cod}(h(x))$.

Remark: These categories are also called \emph{Brandt groupoids}.

====Examples====
Example 1: 

====Basic results====


====Properties====
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

^[[Classtype]]                        |first-order class |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &2\\
  f(3)= &3\\
  f(4)= &7\\
  f(5)= &9\\
  f(6)= &16\\
  f(7)= &22\\
  f(8)= &42\\
  f(9)= &57\\
  f(10)= &90\\
\end{array}$

http://oeis.org/A140189

====Subclasses====
[[Groups]]

====Superclasses====
[[Categories]]

====References====

[(Ln19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]

Trace: • priestley_spaces • ordered_groups • complete_semilattices • join-semidistributive_lattices • polrims • ordered_fields • lie_algebras • closure_algebras • linear_logic_algebras • separation_algebras