Generic Limit Functorial Models and Toposes
by
Cyrus F. Nourani
METAAI and UCSB

Defintion A functor V: A --> X creates limits for a functor F: J --> A if

1. to every limiting cone tt: x --> VF in X there is exactly one pair <a, s > consisting of an object a in A with Va=x and a cone s: a --> F with Vs = tt and if moreover
2. This cone s:a --> F is a limiting cone in A.

Define a functor F: Lw1, B --> Set to be the generic functor defined in [1].

Define a functor V: D<A, G> --> Lw1, B by universal embedding from the diagram functions.

Theorem V Creates a limit for F.

To prove it we have to review some theorems we had proved in our earlier papers.

The papers definitions and theorems put forth topological properties for the infinite language category L1, B, to create limits with generic sets. i-genericity is a topological property defined on an infinitary language, L1, B.

Next, we review what the functor V: D<A, G> --> Lw1, B. The functor V is from a model definable in D<A, G> - the category of models definable with generalized diagram D<A, G> presented by this author in [4], to the limit created from the functor from the Infinitary language category L1, B too Set. D<A, G> is similar to the diagrams defining generic models for set theory. The sets affected are the set of formulas defined by the definition of the fragment L1, B.

Lemma The model defined from the G-diagram, d<F, M> is initial in D<A, G>.

VF is defined by Lw1, B --> D<A, G>.

The Theorem's proof

V creates limits means V defines limits for functors F whose composition VF already has a limit.

VF is defined by L tt fromdefinition 2.1 is an S-indexed family of arrows with vertex in VF and a base set s in OP S. The cone s: a --> F is a limiting cone in Lw1, B Since a G-diagram instance is defined by a unique function set, a generic model is defined by the created limit.

References

[1] Nourani, C.F., "Functorial Model Theory and Infinite Language Categories, " ASL, SF, January 1995. Bulletin ASL, Vol.2, Number 4., December 1996.

[2] Kiesler, H.J. Model Theory For Infinitary Logic, North Holland, Amsterdam, 1971.

[3] McLane, S. Categories For the Working Mathematician, GTM, Springer-Verlag, Berlin-NY-Heildeberg, 1971.

[4] Nourani, C.F, ``Functorial Model Theory, Generic Functors and Sets'', January 16, 1995, International Congress, Logic, Methoodology, and Philosphy of Science, Florence, Italy, August 1995.

Abstract at the author's name

Date received: April 6, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-04.