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functor [2016/04/09 14:29]
nikolaj
functor [2016/04/09 14:31] (current)
nikolaj
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 1. There is not only a collection of special elements $1_a,​1_b,​1_c,​\dots$,​ but also, for each ordered pair of those (such as $\langle 1_a,​1_c\rangle$) there is a whole new collection of elements that's also in ${\bf C}$. 1. There is not only a collection of special elements $1_a,​1_b,​1_c,​\dots$,​ but also, for each ordered pair of those (such as $\langle 1_a,​1_c\rangle$) there is a whole new collection of elements that's also in ${\bf C}$.
  
-2. Each element knows of two other elements. ​(I.e. there is a "domain" ​and "codomain" ​function and these assignments should be obvious form the construction above.)+2. Each element knows of two other elements. I.e. there is a domain and codomain function and these assignments should be obvious form the construction above.
  
-3. There is a "​non-total monoid"​ $\circ$, with the special elements as it's units. ​(It's like a monoid, except it's only partially defined, e.g. $1_a\circ 1_b$ only has a value if $a=b$.)+3. There is a "​non-total monoid"​ $\circ$, with the special elements as it's units. It's like a monoid, except it'​s ​generally ​only partially defined, ​where the domain and codomain function tell you which concatenations make sense (e.g. $1_a\circ 1_b$ only has a value if $a=b$).
  
 A functor is a function that respects $\circ$ in the sense of a monoid-homomorphism. ​ A functor is a function that respects $\circ$ in the sense of a monoid-homomorphism. ​
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