1) 'Les avantages du Kimchi sont nombreux et évidents,' writes Grothendieck his essay on his favorite Korean dish, written at Les Aumettes on 15 October 1983, the same year when he began writing Pursuing Stacks. It is, at its essence, a meaningful perspective on fermentation and preservation. The original scan of the recipe was communicated by Johanna Grothendieck and can be found here. Equally important is the significance attributed to the cultural foundation of the food. He also encourages patience during the whole process of cutting the vegetables (and later fruits in what he called Kimchis sucrés), cleaning the pots, environmental factors, waiting during fermentation, careful preservation, and so on.
An English translation has been done by crowdsourcing and can be found here.
2) While I have never made or eaten Kimchi, I am, however, fascinated by a different culture, namely that of coffee. On a contrasting level of stooping, I have written a brief essay on the cultural chain of coffee production. It can be found here. There is no recipe for good coffee therein, but I remark on the careful journey of a seed to processing and then to careful and different methods of roasting. I believe there must be more precise and professional accounts of the topics in my essay, but my intention was to provide a commentary on the 'artistic' chain of coffee production.
3) If you do not care about either Kimchi or Coffee, then let there be an $\infty$-category (or quasi-category), and what follows is a (turbulent) definition of stable $\infty$-category. An $\infty$-category is called pointed if it has a zero object. The following is an equivalent definition
- The $\infty$-category has an initial object 0.
- The $\infty$-category has a final object 1.
- $0 \simeq 1$
In a pointed category, a triangle is called a fiber sequence if it is a pullback, and a cofiber sequence if it is a pushout. Now, let there be a pointed $\infty$-category. And let a morphism $f: X \to Y$, then we have the following definition of fiber and cofiber (which are just limits in a category). A fiber of $f$ is a fiber sequence (a pullback square)
so $W= fib(f)$. With its dual language, a cofiber of $g$ is a cofiber sequence (a pushout square)
so $Z=cofib(f)$.
An $\infty$-category $\mathcal{C}$ is called stable if the following is satisfied
- It is pointed.
- Every morphism in $\mathcal{C}$ admits a fiber and a cofiber.
- A triangle in $\mathcal{C}$ is a fiber sequence if and only if it is a cofiber sequence. (Basically, a commutative diagram is a pushout square if and only if it is a pullout square.)
The last point (3) in the above definition, which is the coincidence of fiber sequences and cofiber sequences, contributes to the invertibility of the suspension function $\Sigma$ and loop functor $\Omega$. Any exact triangle can thus be rotated using the invertibility. (Well, these functors are self-equivalences of the homotopy category of the $\infty$-category, which is a triangulated category and hence, exact triangles and $\Sigma$ become the translation functor.)
4) But why do we care about a stable $\infty$-category? We have seen previously in The Quill (see here) that there is no canonical functorial way to define (co)-limits in a triangulated category. This poses a problem of gluing, as mentioned by Lurie. For having this feature, we have to develop the stable $\infty$-category for which the homotopy category carries the triangulated structures where the distinguished triangles are the (co)fiber sequences. A prime example developed by Lurie is the $\infty$-category of Spectra (which is an interesting discussion in its own right). It is interesting to study the motivation behind the development of the theory of stable $\infty$-categories. We will discuss them in due course.
