Mark Spezzano
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!