Left oplax Kan extensions turn relative 2-monads into pseudomonads with the same colax algebras, producing the weak ultracompletion pseudomonad for ultracategories.
Towards a General Theory of Dependent Sums
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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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Ultracategories via Kan extensions of relative monads
Left oplax Kan extensions turn relative 2-monads into pseudomonads with the same colax algebras, producing the weak ultracompletion pseudomonad for ultracategories.