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Towards a General Theory of Dependent Sums

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arxiv 2404.08139 v1 pith:SCRT43WW submitted 2024-04-11 math.CT math.RA

Peter Bonart This is my paper · ORCID
classification math.CTmath.RA
keywords dependentexamplessumsaddersmanyadderaddingbranches
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abstract

We introduce dependent adders. A dependent adder $A$ has for every $x \in A$ a way of adding together $x$ many elements of $A$. We provide examples from many disparate branches of mathematics. Examples include the field with one element $\mathbb{F}_1$, the real numbers with integrals as sums, the category of categories with oplax colimits as sums. We also consider modules over dependent adders and provide examples.

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  1. Ultracategories via Kan extensions of relative monads

    math.CT 2025-06 conditional novelty 7.0 of 10

    Left oplax Kan extensions turn relative 2-monads into pseudomonads with the same colax algebras, producing the weak ultracompletion pseudomonad for ultracategories.

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