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Topic: Toward a Formal, Machine-Parsable Definition of a Category
Replies: 8   Last Post: Nov 6, 2012 8:39 AM

 Messages: [ Previous | Next ]
 Ki Song Posts: 549 Registered: 9/19/09
Re: Toward a Formal, Machine-Parsable Definition of a Category
Posted: Nov 6, 2012 1:01 AM

On Monday, November 5, 2012 10:46:48 PM UTC-5, 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

Date Subject Author
11/5/12 Aatu Koskensilta
11/5/12 Dan Christensen
11/5/12 Rotwang
11/6/12 Dan Christensen
11/6/12 Rotwang
11/6/12 Ki Song
11/6/12 Frederick Williams
11/6/12 Ki Song