Let A be a class of partial orders and B be a class of complete embeddings between elements of A closed under composition.Then (A,B) is a category whose objects are elements of A and whose arrows are elements of B. Moreover (A∩Vδ,B∩Vδ) is a partial order. Depending on the nature of A and B this can be an interesting or trivial partial order. If A is the class of all posets and B is the class of all complete embeddings and δ is limit (A∩Vδ,B∩Vδ) is a trivial partial order since all elements of this partial order are compatible.

We shall study the case in which A is the class of stationary set preserving (semiproper, proper) posets, and B is the class of complete embeddings between elements of A with a stationary set preserving (semiproper, proper) quotient.

We show that if δ has some degree of largeness which depends on the choice of the category (A,B), (A∩Vδ,B∩Vδ) is a STATIONARY SET PRESERVING partial order which collapses δ to become ℵ2 but it should NEVER be a proper one and can be a semiproper one only if MM++ holds in the ground model.

Finally we briefly outline how these partial orders can be of use to study absoluteness results for the theory of the Chang model for sets of size ℵ1. However this will be the subject of a future talk.