Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
From what I can gather, sheaves on (some) Vect (with some Grothendieck topology I don’t recall) constitute a standard model for generalized smooth spaces. I still recall my alarm when I first learned that this produces a synthetic differential geometry equivalent to that obtained by taking sheaves on the category of cartesian spaces and all smooth maps between them, which I believe is called Cart when equipped with (some Grothendieck topology I don’t recall). Resolving my surprise concerning this identification was formative in developing my intuition for the various flavors of completion 2-functors such as Karoubi envelopes and Cauchy completion in the enriched context (and Yoneda etc).
Forgive me if a flaw in my understanding has resulted in a silly question, but I am quite curious what would be obtained by extending this recipe along the analogy/abstraction [R : R-Bimod] :: [V : V-Prof]. In short: what are “spaces locally modelled on enriched categories” in the sense that tangent spaces are (enriched) categories, tensor fields are diagrams of profunctors indexed by the shape ∫ of the base, etc.
Does anyone know if this has been addressed in the literature? I would love to read more! Thank you
sheaves on (some) $\mathbf{Vect}$ (with some Grothendieck topology I don’t recall) constitute a standard model for generalized smooth spaces.
Do you have a pointer to what you have in mind here?
the category of cartesian spaces and all smooth maps between them, which I believe is called $\mathbf{Cart}$
Yes. I had turned attention to this category as a site for smooth spaces in Def. 1.2.197 of dcct (p. 144) and the idea stuck (previous authors had insisted on using categories of open subsets of Euclidean spaces, instead).
Hmm. After struggling to track down where I got the idea about Vect serving as a nice site for smooth spaces, I wonder if I got confused reading an older version of CartSp or smooth set. Maybe my alarm was actually good sensibility all along, haha.
Thank you for the reference. I will keep chewing on the relationship between Vect and Prof; right now I’m toying with slices $Prof_{/1}$ as analogs of $Vect_{/k[1]} \simeq Aff$.
1 to 5 of 5