University of Birmingham > Talks@bham > Lab Lunch > The coproduct of frames as encoding d-frame structure

The coproduct of frames as encoding d-frame structure

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If you have a question about this talk, please contact Dr Steve Vickers.

Bitopological spaces have as pointfree representations structures called d-frames. This is in virtue of a Stone-type contravariant adjunction between the two categories at play.

D-frames are quadruples (L_1, L_2, con, tot) where L_1,L_2 are frames and con, tot are subsets of their product satisfying certain axioms. The frames L_1 and L_2 represent frames of opens of two topologies imposed on the same set. The subsets con and tot represent respectively the disjoint pairs and those whose union covers the whole space.

We will see how the coproduct (L_1)+(L_2) of two frames L_1,L_2 encodes information on the possible con and tot subsets of (L_1)x(L_2) satisfying the d-frame axioms. In particular cons and tots may be seen as certain sublocales of the coproduct.

D-frames may also be seen as presentations of frames, where the first two components are generators and the last two are relations. Given any pair of frames L_1,L_2 we will show how to construct a d-frame such that it presents a finitary substructure of the assembly of the coproduct (L_1)+(L_2). This finitary assembly also enjoys a universal property similar to that of the assembly.

This talk is part of the Lab Lunch series.

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