Difference between revisions of "Coalgebra"

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(Created page with "Let <math>\mathbf{C}</math> be a category and <math>F : \mathbf{C} \to \mathbf{C}</math> an endofunctor on <math>\mathbf{C}</math>. An ''<math>F</math>-coalgebra'' is a morphi...")
 
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Let <math>\mathbf{C}</math> be a category and <math>F : \mathbf{C} \to \mathbf{C}</math> an endofunctor on <math>\mathbf{C}</math>. An ''<math>F</math>-coalgebra'' is a morphism <math>c : X \to F X</math> in <math>\mathbf{C}</math>.
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Let <math>\mathbf{C}</math> be a category and <math>F : \mathbf{C} \to \mathbf{C}</math> an endofunctor on <math>\mathbf{C}</math>. An ''<math>F</math>-coalgebra'' is a morphism <math>c : X \to F X</math> in <math>\mathbf{C}</math>. A ''homomorphism'' from a coalgebra <math>c : X \to F X</math> to <math>d : Y \to F Y</math> is a morphism <math>f : X \to Y</math> such that <math>F f \circ c = d \circ f</math>.
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<theorem id="coalg-cat">'''%c''' &nbsp;
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  <math>F</math>-Coalgebras and their homomorphisms form a category <math>\mathrm{Coalg}(F)</math>.
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</theorem>

Revision as of 22:54, 11 February 2015

Let [math]\mathbf{C}[/math] be a category and [math]F : \mathbf{C} \to \mathbf{C}[/math] an endofunctor on [math]\mathbf{C}[/math]. An [math]F[/math]-coalgebra is a morphism [math]c : X \to F X[/math] in [math]\mathbf{C}[/math]. A homomorphism from a coalgebra [math]c : X \to F X[/math] to [math]d : Y \to F Y[/math] is a morphism [math]f : X \to Y[/math] such that [math]F f \circ c = d \circ f[/math].

<theorem id="coalg-cat">%c  

 [math]F[/math]-Coalgebras and their homomorphisms form a category [math]\mathrm{Coalg}(F)[/math].

</theorem>