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Compositories and gleaves

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Cecilia Flori and Tobias Fritz

Sheaves are objects of a local nature: a global section is determined by
how it looks locally. Hence, a sheaf cannot describe mathematical
structures which contain global or nonlocal geometric information. To fill
this gap, we introduce the theory of ``gleaves'', which are presheaves
equipped with an additional ``gluing operation'' of compatible pairs of
local sections. This generalizes the *conditional product* structures
of Dawid and Studeny, which correspond to gleaves on distributive
lattices. Our examples include the gleaf of metric spaces and the gleaf of
joint probability distributions. A result of Johnstone shows that a
category of gleaves can have a subobject classifier despite not being
cartesian closed.

Gleaves over the simplex category $\Delta$, which we call compositories,
can be interpreted as a new kind of higher category in which the
composition of an m-morphism and an n-morphism along a common
k-morphism face results in an (m+n-k)-morphism. The distinctive
feature of this composition operation is that the original morphisms can
be recovered from the composite morphism as initial and final faces.
Examples of compositories include nerves of categories and compositories
of higher spans.

Keywords:
sheaf theory, uniqueness of gluing, nerve of a category, higher span,
conditional product distribution, Lawvere metric space, relational
database

2010 MSC:
Primary 18F20, Secondary 18A99, 18F10

*Theory and Applications of Categories,*
Vol. 31, 2016,
No. 33, pp 928-988.

Published 2016-10-23.

http://www.tac.mta.ca/tac/volumes/31/33/31-33.pdf

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