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at.algebraic topology – Groupoids, cubical units, and Segway situations retort

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at.algebraic topology – Groupoids, cubical units, and Segway situations

There is a fairly good presentation of the class of diminutive classes as a pullback in CAT (graze the exposition in Mellie’s “Segal condition proper computation effects”, as an 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 an entire 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 right down to graphs follows from Weber’s nerve theorem).

It definitely feels as if there ought to breathe an identical development 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 doubtless breathe derived.

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