Re: Full and faithful

Re: Full and faithful

[Ross Street]

Perhaps "representably fully faithful" is a good name.

In Cat, they are the fully faithful functors.

In V-Cat they may not be fully faithful V-functors: see

(with R.F.C. Walters) Yoneda structures on 2-categories, J. Algebra
50 (1978) 350-379

Also see

(with A. Carboni, S. Johnson and D. Verity) Modulated bicategories,
J. Pure Appl. Algebra 94 (1994) 229-282

---Ross

On 31/03/2007, at 3:44 AM, Robin Houston wrote:

> A functor F: C -> D is full and faithful just when, for all
> categories X and functors G, H: X -> C, the whiskering action of F
> induces a bijection between [G, H] and [FG, FH]  (where [G, H]
> denotes the set of natural transformations from G to H).
>
> Clearly this formulation makes sense in any bicategory. Is there a
> name for 1-cells with this property?

Full and faithful

[Robin Houston]

A functor F: C -> D is full and faithful just when, for all
categories X and functors G, H: X -> C, the whiskering action of F
induces a bijection between [G, H] and [FG, FH]  (where [G, H]
denotes the set of natural transformations from G to H).

Clearly this formulation makes sense in any bicategory. Is there a
name for 1-cells with this property?

Thanks!

Robin