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 July 30th, 2019, 05:01 AM #1 Newbie   Joined: Jul 2019 From: India Posts: 1 Thanks: 0 Category Theory - Monad Hello, I am a programmer and encountered Monad as functional programming pattern. I want to deeply learn the concept so I learned some points from category theory which can explain me Monad from math's perspective. I learned composition, functor, natural transformation and adjunction. I learned that adjunction creates monad-comonad pair. Suppose there is category C and D. Each x from C maps to Gx via functor $\displaystyle G: C \rightarrow D$ in category D, and each y in D maps to Fy in C via functor $\displaystyle F: D \rightarrow C$ and there is $\displaystyle f : Fy \rightarrow x$ and $\displaystyle g : y \rightarrow Gx$ We have adjunction with $\displaystyle {Hom}_C(Fy, x) \cong {Hom}_D(y, Gx)$. Here, we get $\displaystyle {\eta}_y : y \rightarrow GFy$, $\displaystyle {\eta}_{Gx} : Gx \rightarrow GFGx$, $\displaystyle {\eta}_{GFy} : GFy \rightarrow GFGFy$, $\displaystyle {\eta}_{GFGx} : GFGx \rightarrow GFGFGx$ and $\displaystyle {\epsilon}_x : FGx \rightarrow x$, $\displaystyle {\epsilon}_{Fy} : FGFy \rightarrow Fy$, $\displaystyle {\epsilon}_{FGx} : FGFGx \rightarrow FGx$, $\displaystyle {\epsilon}_{FGFy} : FGFGFy \rightarrow FGFy$ in adjunction such that above hom-set satisfied. Now, $\displaystyle G \circ F$ is a monad and $\displaystyle F \circ G$ is a comonad Monad has two natural transformations $\displaystyle \eta : I \rightarrow GF$ and $\displaystyle \mu : GFGF \rightarrow GF$ and Comonad has below two natural transformations $\displaystyle \epsilon: FG \rightarrow I$ $\displaystyle \delta : FG \rightarrow FGFG$ I come to know that natural transformation $\displaystyle \eta : I \rightarrow GF$ has naturality square like $\displaystyle {\eta}_{Gx} \circ g = GF(g) \circ {\eta}_y$ and $\displaystyle \epsilon : FG \rightarrow I$ has $\displaystyle f \circ {\epsilon}_{Fy} = {\epsilon}_x \circ FG(f)$ That's how I can get first laws for monad and comonad, but I cannot wrap my head around second laws for both. $\displaystyle \mu : GFGF \rightarrow GF$ $\displaystyle \delta: FG\rightarrow FGFG$ How do this above laws emerge from adjunction? Which naturality squares exist for above two laws? It will be a great help as I am stuck only on the last point. Thanks Tags category, monad, theory Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post raul21 Math 0 May 24th, 2014 01:53 AM ipp Abstract Algebra 0 December 8th, 2013 08:39 AM butabi Abstract Algebra 8 September 3rd, 2011 01:52 PM mingcai6172 Abstract Algebra 2 January 7th, 2009 03:09 AM

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