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categories [2010/07/29 15:46]
127.0.0.1 external edit
categories [2016/11/27 09:54] (current)
jipsen
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=====Categories===== =====Categories=====
-% Note: replace "Template" with Name_of_class in previous line 
Abbreviation: **Cat** Abbreviation: **Cat**
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$\langle C,\circ\rangle$ is a (large) [[partial semigroup]] $\langle C,\circ\rangle$ is a (large) [[partial semigroup]]
 +
 +dom amd cod are total unary operations on $C$ such that
$\text{dom}(x)$ is a left unit:  $\text{dom}(x)\circ x=x$ $\text{dom}(x)$ is a left unit:  $\text{dom}(x)\circ x=x$
$\text{cod}(x)$ is a right unit:  $x\circ\text{cod}(x)=x$ $\text{cod}(x)$ is a right unit:  $x\circ\text{cod}(x)=x$
- 
-$\text{dom}(\text{dom}(x))=\text{dom}(x)=\text{cod}(\text{dom}(x))$ 
- 
-$\text{cod}(\text{cod}(x))=\text{cod}(x)=\text{dom}(\text{cod}(x))$ 
if $x\circ y$ exists then $\text{dom}(x\circ y)=\text{dom}(x)$ and $\text{cod}(x\circ y)=\text{cod}(y)$ if $x\circ y$ exists then $\text{dom}(x\circ y)=\text{dom}(x)$ and $\text{cod}(x\circ y)=\text{cod}(y)$
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==Morphisms== ==Morphisms==
Let $\mathbf{C}$ and $\mathbf{D}$ be categories. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a homomorphism: Let $\mathbf{C}$ and $\mathbf{D}$ be categories. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a homomorphism:
-$h(\text{dom}(c))=\text{dom}h(c)$ and+$h(\text{dom}(c))=\text{dom}h(c)$, $h(\text{cod}(c))=\text{cod}h(c)$ and
$h(c\circ d)=h(c) \circ h(d)$ whenever $c\circ d$ is defined. $h(c\circ d)=h(c) \circ h(d)$ whenever $c\circ d$ is defined.
 +
 +Morphisms between categories are called \emph{functors}.
====Examples==== ====Examples====
Example 1: The category of function on sets with composition. Example 1: The category of function on sets with composition.
 +In fact, most of the classes of mathematical structures in this database are categories.
====Basic results==== ====Basic results====
 +$\text{dom}(\text{dom}(x))=\text{dom}(x)=\text{cod}(\text{dom}(x))$
 +
 +$\text{cod}(\text{cod}(x))=\text{cod}(x)=\text{dom}(\text{cod}(x))$
====Properties==== ====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]]                        |(value, see description) [(Ln19xx)] |+^[[Classtype]]                        |many-sorted variety |
^[[Equational theory]]                | | ^[[Equational theory]]                | |
^[[Quasiequational theory]]           | | ^[[Quasiequational theory]]           | |
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$\begin{array}{lr} $\begin{array}{lr}
  f(1)= &1\\   f(1)= &1\\
-  f(2)= &\\ +  f(2)= &3\\ 
-  f(3)= &\\ +  f(3)= &11\\ 
-  f(4)= &\\ +  f(4)= &55\\ 
-  f(5)= &\\ +  f(5)= &329\\ 
-\end{array}$      +  f(6)= &2858\\
-$\begin{array}{lr} +
-  f(6)= &\\+
  f(7)= &\\   f(7)= &\\
  f(8)= &\\   f(8)= &\\
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\end{array}$ \end{array}$
 +http://oeis.org/A125696
====Subclasses==== ====Subclasses====
-  [[...]] subvariety+[[Schroeder categories]] 
 + 
 +[[Closed categories]]
-  [[...]] expansion+[[Compact categories]]
====Superclasses==== ====Superclasses====
-  [[...]] supervariety+[[Partially ordered categories]]
-  [[...]] subreduct+[[Partial semigroups]]