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schroeder_categories [2016/11/27 10:14]
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schroeder_categories [2016/11/28 19:04] (current)
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====Definition==== ====Definition====
-A \emph{Schroeder category} is a [[catergory]] $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$ such that+A \emph{Schroeder category} is an enriched [[category]] $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$
-every morphism is an isomorphism: $\forall x\exists y\ x\circ y=\text{dom}(x)\text{ and }y\circ x=\text{cod}(x)$ +in which every hom-set is a Boolean algebras.
==Morphisms== ==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 $h(\text{cod}(x))=\text{cod}(h(x))$. 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{groupoids}.
====Examples==== ====Examples====
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$\begin{array}{lr} $\begin{array}{lr}
  f(1)= &1\\   f(1)= &1\\
-  f(2)= &1\\ +  f(2)= &\\ 
-  f(3)= &2\\ +  f(3)= &\\ 
-  f(4)= &3\\ +  f(4)= &\\ 
-  f(5)= &7\\ +  f(5)= &\\ 
-  f(6)= &9\\ +  f(6)= &\\ 
-  f(7)= &16\\ +  f(7)= &\\ 
-  f(8)= &22\\ +  f(8)= &\\ 
-  f(9)= &42\\ +  f(9)= &\\ 
-  f(10)= &57\\+  f(10)= &\\
\end{array}$ \end{array}$
 +
====Subclasses==== ====Subclasses====
-  [[...]] subvariety +[[...]]
- +
-  [[...]] expansion +
====Superclasses==== ====Superclasses====
-  [[...]] supervariety +[[Categories]]
- +
-  [[...]] subreduct +
====References==== ====References====