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Added another reference.
I was chatting with Robin Cockett yesterday at SYCO1. In a talk Robin claims to be after
The algebraic/categorical foundations for differential calculus and differential geometry.
It would be good to see how this approach compares with differential cohesive HoTT.
Added another paper from this approach
in this paper we are interested in determining how to define differential forms, their exterior derivative, and the resulting cochain complex of de Rham in an arbitrary tangent category.However, to do so requires a close inspection of the nature of differential forms. This inspection reveals an interesting structure, a simplicial object of sector forms, of which de Rham cohomology can be seen as a simple consequence.
Re #1, that’s a remarkable claim to make after some decades of SDG.
I don’t have a feeling yet for what these developments are really aiming for.
Re the latter sentence of #3, it seems that this difficulty is mutually felt.
I wonder if
could help with translation. It is argued there that to give a tangent structure on a category $M$ is to give a functor for (some subcategory) of the category of Weil algebras (in the Artin algebra sense) to the endofunctor category $[M, M]$.
Hi, I work with tangent bundle categories and would like to fill out nLab page - maybe this could help bridge the gap. The tangent bundle category approach mostly grew out of studying categorical models of Linear Logic: once you start thinking about the “linear approximation” of a proof $!(A) \to B$, you end up with something that models differential calculus (differential linear logic, the differential lambda calculus, differential and cartesian differential categories). It’s fairly surprising that this leads you back to synthetic differential geometry/the Weil functor approach to differential geometry.
I work with tangent bundle categories and would like to fill out nLab page
Please feel invited to! Just hit “edit” at the bottom of the page and get going. Syntax is fairly straightforward, let us know if you have questions. And at first go, not to worry about anything coming out right or not, we will help with the formatting.
once you start thinking about the “linear approximation” of a proof $!(A) \to B$, you end up with something that models differential calculus
Around here we think of this in terms of dependent linear type theory with categorical semantics in indexed monoidal (∞,1)-categories, see at Exponential modality and Fock space.
The canonical such semantic model are tangent (∞,1)-categories in the sense of Lurie. Notice that this is terminology introduced indepdently (and I think earlier) than the tangent categories that you are referring, to – but luckily the concepts are actuall very close to each other.
One way to formalize this close conceptual relation to is observe that tangent (∞,1)-categories are “infinitesimal cohesive (∞,1)-toposes”, see there.
Thanks!
Your last pullback diagram currently is lacking its left vertical arrow.
Thanks for catching that! Also, I’m very sorry about the multiple posts there, I’m still getting the hang of this.
BTW, regarding the naming issue in the first lines of the entry: the logical term for what this entry describes would have been “category of spaces with tangent bundles”, or maybe less precisely but more catchy: “category of tangent bundles”. The other concept otoh, does logically deserve to be called “tangent category”.
I think the naming convention came from “a category equipped with a tangent structure is a tangent category”, which is similar to how you might see a monoidal category or restriction category defined. I can see how “the tangent category of $\mathbb{X}$” could be more intuitive.
I think there has been some success connecting the two definitions, based on the BJORT paper (at least at the level of functor calculus).
I don’t get the feeling that you picked up the point I was making on terminology. But it’s not important. And too late anyways! :-)
The other concept otoh, does logically deserve to be called “tangent category”.
Actually, when I spoke to Eric Finster about this at the modal HoTT workshop, he suggested that the $(\infty,1)$-topos of parametrized spectra (the “tangent $(\infty,1)$-category” of $\infty Gpd$) ought rather to be considered analogous to the formal discs $\mathbb{D}$ of infinitesimals in SDG, so that the actual “tangent bundle” of a topos would be obtained by exponentiating with it.
I see. Right, that makes good sense.
Re #15, is that related to the exponential/logarithm talk at Goodwillie calculus, from around
the Goodwillie tower is the homotopy theoretic analog of logarithmic expansion, rather than of Taylor series ?
I dunno. I’m never sure how seriously to take these analogies.
That comment is from a paper by Eric’s supervisor, Greg Arone, so maybe he takes it seriously.
Hi, my collaborator Jonathan Gallagher will also be contributing to tangent bundle categories. He’s liked Felix Wellen’s thesis, and has thought about how it relates to tangent bundle categories.
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