Added the reference
* Geoff Cruttwell and Jean-Simon Lemay, *Tangent categories as a bridge between differential geometry and algebraic geometry*, 2023. (arXiv:2301.05542)

as noticed here, this entry was not rendering, instead showing an (unintelligible) error message.

I have fixed it now: The problem was indented `tikzcd`

diagrams.

(See the current *Sandbox* for a minimal (non-)working example of this bug).

Namely, in general our `tikz`

-functionality is, currently, an alien add-on which does not play along with the ambient Instiki code.

The only way (as far as I am aware) to display `tikzcd`

in $n$Lab pages is as stand-alone environments whose lines `\begin{tikzcd}`

and `\end{tikzcd}`

are aligned to the abosolute left.

If coded like so, the `tikzcd`

-content displays, but of course it now breaks all numered item lists that it may be sitting in. I have worked around this in the entry by making a bullet-item list with manual numbering added.

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.

]]>That comment is from a paper by Eric’s supervisor, Greg Arone, so maybe he takes it seriously.

]]>I dunno. I’m never sure how seriously to take these analogies.

]]>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 added the actegory definition of a tangent category. To simplify the construction, I restricted to the case where tangent categories have negatives.

]]>I see. Right, that makes good sense.

]]>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 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! :-)

]]>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).

]]>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”.

]]>Thanks for catching that! Also, I’m very sorry about the multiple posts there, I’m still getting the hang of this.

]]>Thanks!

Your last pullback diagram currently is lacking its left vertical arrow.

]]>I’ve added the classical definition of a tangent category. I will add the actegory and enriched versions shortly, but I have included some references related to the two approaches.

]]>I’ve added the classical definition of a tangent category. I will add the actegory and enriched versions shortly, but I have included some references related to the two approaches.

]]>I’ve added the classical definition of a tangent category. I will add the actegory and enriched versions shortly, but I have included some references related to the two approaches.

]]>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.

]]>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.

]]>Re the latter sentence of #3, it seems that this difficulty is mutually felt.

I wonder if

- Poon Leung,
*Classifying tangent structures using Weil algebras*, Theory and Applications of Categories, 32(9):286–337, 2017, (tac)

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 $[M, M]$.

]]>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.

]]>Added another paper from this approach

- Geoff Cruttwell, Rory Lucyshyn-Wright,
*A simplicial foundation for differential and sector forms in tangent categories*, (arXiv:1606.09080)

]]>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.

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.

]]>