Dear Ross, I think you meant the *product* completion Fam(D*)*. If you use the coproduct completion, then you need to use discrete *opfibrations* and covariant Set-valued functors. There are some details in the appendix to our joint paper “The formal theory of monads II”. Steve.
On 18 Mar 2015, at 11:47 am, Ross Street <ross.street@mq.edu.au> wrote:
Dear All
David Leduc’s statement: "A partial functor from C to D is given by a subcategory S of C and a functor from S to D.”
If this is taken as given then today I don’t want to add to what people have said.
But there is another notion of partial functor that I learnt from Brian Day who learnt it from Bill Lawvere. If we are to replace 2 in Set by Set in Cat, then characteristic maps C —> 2 should become presheaves C^op —> Set, so injective functions S—> C should become discrete fibrations E —> C. Then a partial map from C to D would be a span C <— E —> D with C <— E a discrete fibration and E —> D any functor. The partial map classifier is then the coproduct completion FamD of D. Add size restrictions, to taste.
Ross
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