|Location:||MSRI: Baker Board Room|
Decomposition spaces (Galvez-Carrillo, Kock, Tonks), or 2-Segal spaces (Dyckerhoff, Kapranov), are simplicial spaces which can be thought of as categories enriched in Span. In this talk we discuss connections between decomposition spaces and operadic categories. Carlier provides a zig-zag between the two settings, by showing that each of Schmitt's hereditary species determines both a monoidal decomposition space and an operadic category. Garner, Kock, and Weber provide another connection, by characterizing unary operadic categories (that is, those operadic categories where all objects have cardinality 1) as simplicial sets whose (top) decalage is 1-Segal. As the decalage of a 2-Segal set is 1-Segal, this exhibits discrete decomposition spaces as a special kind of unary operadic category.
We introduce (op)fibrations between operadic categories and the related Grothendieck's construction. We will show how to use it to construct new operadic categories from old ones. If time permits, we mention applications to infinity operads.
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