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Ki Song
Posts:
394
Registered:
9/19/09


Re: Toward a Formal, MachineParsable Definition of a Category
Posted:
Nov 6, 2012 1:01 AM


On Monday, November 5, 2012 10:46:48 PM UTC5, Dan Christensen wrote: > On Nov 5, 7:03 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > > > Dan Christensen <Dan_Christen...@sympatico.ca> writes: > > > > Agreed. But leaving aside your own examples for now, in the example I > > > > just posed to Rotwang and Aatu, you then agree that it doesn't matter > > > > whether g o f = h1 or h2  I should just pick one. If I chose g o f = > > > > h1, then, even though h2 is, in a sense, also a composition of f and > > > > g, we would have g o f =/= h2. > > > > > > In what sense is h2 also a composition of f and g? > > > > In the sense that dom(h2)=dom(f) and cod(h2)=cod(g).
That does not mean that h2 is a composition of f and g.
> > > > > All you know is > > > that h2 has suitable domain and codomain. > > > > That should be enough.
No.
> > > > "For each ordered triple of objects A, B, C in category [curly] C, > > there is a law of composition: If f:A>B and g:B>C, then the > > composite of f and g is a morphism gf:A>C."
It says there is an arrow gf, it doesn't say that every arrow from A to C is equal to gf.
> > > > Source: "Category Theory," WikiBooks, http://en.wikibooks.org/wiki/Category_Theory/Categories#Definition > > > > Doesn't that suggest that ANY morphism mapping A to C (in the > > defintion here) would do for the composite of f and g?
No.
> Did WikiBooks > > get it wrong? >
You got it wrong. > > > > > > Whether it's the composition > > > of f and g depends on the f and g in question, and the relevant notion > > > of composition given by the category at issue. > > > > > > > I'm just going by the above definition, and it suggests that the > > domain and codomain are the ONLY relevant criteria. A problem arises, > > of course, when there are multiple, distinct morphisms from A to C  > > which one to pick? It's beginning to look to me like the choice is > > completely arbitrary. > > > > Can you cite any authoritative source to contrary? Online would be > > nice. > > > > > > > > I can live with that. It just means that, in some cases g o f is NOT > > > > the only composition of f and g. > > > > > > Whether h2 or h1 is the composition of f and g is determined by the > > > category. > > > > Again, the definition above suggests that the only determining > > criteria are the domain and codomain. > >
No, it does not. How in the world can you get a reading like that? > > > > > Again, do note the (partial binary) operation of composition > > > is part of the data that defines a category. > > > > > > > It seems to me that would it work perfectly fine to arbitrary select > > as the composite any morphism with the required domain and codoemain. > > This would at least be consistent with all the definitions we have > > looked at. > > > > Dan > > Download my DC Proof 2.0 software a http://www.dcproof.com



