[Haskell-cafe] Re: Category Theory woes
L Spice
HaskellJade.5.JadeNB at spamgourmet.com
Wed Feb 10 00:13:54 EST 2010
Mark Spezzano <mark.spezzano <at> chariot.net.au> writes:
> Does anyone know what Hom stands for?
'Hom' stands for 'homomorphism' --a way of changing (morphism)
between two structures while keeping some information the same (homo-).
Any algebra text will define morphisms aplenty --homomorphisms,
epimorphisms, monomorphisms, and the like. These are maps on groups
that preserve group operations (or on rings that preserve ring operations,
etc.)
In a topology text, you will find information on what are called
continuous functions; they're morphisms too, in disguise. You can find a
thinner disguise when you look at continuously invertible continuous
functions, which are called homeomorphisms. If you proceed to differential
geometry, you'll see smooth maps --they're morphisms too, and the
invertible ones are called diffeomorphisms.
This-morphisms, that-morphisms --if you're trying to come up with a
general theory that describes all of them, it's natural just to call them
'morphisms'; but, as with the word 'colonel', the word and the symbol come to
us via different routes, so that 'Hom(omorphism)' survives instead as the
abbreviation. The crucial point in learning category theory is the realisation
that, despite all the fancy terminology, it is at heart nothing but a way of
talking about groups, rings, topological spaces, partial orders, etc.
--all at once, so no wonder it seems abstract!
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