Re: initial algebra question

Re: initial algebra question

[Steve Vickers]

Dear Paul,

I proved a "topical" version of this as Propn 2.3.7 in my "Topical
Categories of Domains" (1999).

"Topical" here means working in the 2-category of Grothendieck
toposes and geometric morphisms instead of that of categories and
functors. Instead of objects of a category and morphisms between
them, it deals with points of a topos and natural transformations
between them. (Note that I use the term "F-structures" instead of "F-
algebras".)

In this setting there are some subtleties of interpretation. An
initial F-structure is defined as a point of the classifying topos [F-
struct] for F-structures that is initial amongst all the generalized
points - making [F-struct] a local topos. Nonetheless, the argument
is essentially one that you might use with categories and functors.

I remarked that my results were familiar from the category context as
set out in Freyd's 1991 paper "Algebraically complete categories". I
cannot remember if your result on FG-algebras and GF-algebras was in
Freyd.

All the best,

Steve.


On 28 Jan 2009, at 19:31, Paul Levy wrote:

> Does anybody know a reference for the following (very easy) result?
>
> Let C and D be categories, and let F:C-->D and G:D-->C be functors.
>
> If (c,theta) is an initial algebra for GF, then (Fc, F theta) is an
> initial algebra for FG.
>
> thanks,
> Paul
>
>
>

Re: initial algebra question

[Adam Eppendahl]

> Does anybody know a reference for the following (very easy) result?
> Let C and D be categories, and let F:C-->D and G:D-->C be functors.
> If (c,theta) is an initial algebra for GF, then (Fc, F theta) is an
> initial algebra for FG.

It is in Section 5 of

Peter Freyd
Remarks on Algebraically Compact Categories, LMS LNS 177, 1992.

(modulo initial invariant = initial algebra).

The full dinaturality of initial algebra delivery (as a diagram of
functors) is in Section 4 of

Adam Eppendahl
Coalgebra-to-algebra Morphisms, ENTCS 29, 1999.

where it is seen to follow from the lemma:

If p is a coalgebra for GF and s is an algebra for FG, then morphisms
from Fp to s correspond one-for-one to morphisms from p to Gs (even
without an adjunction between G and F).

Adam Eppendahl

initial algebra question

[Paul Levy]

Does anybody know a reference for the following (very easy) result?

Let C and D be categories, and let F:C-->D and G:D-->C be functors.

If (c,theta) is an initial algebra for GF, then (Fc, F theta) is an initial algebra for FG.

thanks,
Paul