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Stream: theory: category theory

Topic: equipments of algebras

Christian Williams (Dec 10 2023 at 02:20):

Today I thought of the following conjecture; I'll be drawing and writing it out. Let me know of any thoughts.

Let $\mathscr{T}:\mathbb{C\to C}$ be a pseudomonad on a bifibrant double category, a.k.a. equipment; such a double functor is a transversal morphism in the "triple category" (metalogic) of equipments. Then:

The equipment of free algebras $\mathrm{Kl}(\mathscr{T})$ is (the base-category-restriction of) the collage of the horizontal conjoint $\mathbb{C}(1,\mathscr{T})$.

The equipment of algebras $\mathrm{Alg}(\mathscr{T})$ is the tabulation of the vertical companion $\underline{\mathbb{C}}(\mathscr{T},1)$.