Session S27 - Categories and Topology

Wednesday, July 21, 17:00 ~ 17:30 UTC-3

Exponentiable Inclusions: Quantaloids and Ringoids

Susan Niefield

Union College, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In earlier work, we showed that the inclusion of a subobject is exponentiable if and only if it is locally closed for five categories; namely, spaces (1978), locales (1980), toposes (1980), small categories (2000), and posets (2001). In 2012, we introduced the notions of locally closed inclusions and Artin-Wraith glueing in double categories. With appropriate assumptions, we showed that locally closed inclusions are exponentiable, and constructed the exponentials via the glueing condition. This provided a single theorem establishing the exponentiability of locally closed inclusions that applied to the five categories mentioned above.

In this talk, we will show that if $\cal V$ is a cocomplete symmetric monoidal category, then categories enriched in $\cal V$ are the objects of a double category with the appropriate glueing properties, and hence, we obtain the exponentiability of locally closed inclusions of $\cal V$-categories. Furthermore, we will see that the locally closed condition is also necessary when $\cal V$ is monadic over the category of sets. Thus, we obtain a characterization of the exponentiable inclusions of quantaloids (respectively, ringoids) when $\cal V$ is the category of suplattices (respectively, abelian groups) analogous to that of the five earlier cases.

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