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at.algebraic topology – Groupoids, cubical units, and Segal circumstances retort

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at.algebraic topology – Groupoids, cubical units, and Segal circumstances

There is a slightly good presentation of the class of diminutive classes as a pullback in CAT (graze the exposition in Mellies’s “Segal condition proper computation effects”, for example).

In the class of graphs, level to that the subcategory of objects representing the “paths of length n” functor is dense (convene this Path), so the class of graphs is a whole subcategory of presheaves on Path. too level to there’s a bijective on objects functor from Path to the simplex class, giving a monadic functor from simplicial units to presheaves on Path. The pullback in Cat is exactly the whole subcategory of simplicial units satisfying the Segal situation (modacity of the unmindful functor all the way down to graphs follows from Weber’s nerve theorem).

It definitely feels as if there ought to breathe an analogous building of the class of groupoids utilizing cubical units (with connection) within the literature, however I haven’t been capable of path it down. The closest is the nLab’s web page, the place the Segal situation can possible breathe derived.

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