"The total chain complex is, under the Dold-Kan correspondence, equivalent to the diagonal of a bisimplicial set – see Eilenberg-Zilber theorem. As discussed at bisimplicial set, this is weakly homotopy equivalent to the total simplicial set of a bisimplicial set."

Would it be that the total chain complex is **exactly** the total(i.e. codiagonal or $\bar{W}$) simplicial abelian group of the bisimplicial abelian group under Dold-Kan correspondence? Sorry if I am wrong. ]]>

At holomorphic function, a function between complex manifolds is defined as holomorphic if it is complex-differentiable. Given that underlying a complex manifold is a smooth manifold, ought we use the smooth structure to define a holomorphic function between complex manifolds as a smooth function annihilated by the antiholomorphic differential?

From my understanding, complex-differentiability is pertient when we speak of functions, say, from ${\mathbb{C}}$ to ${\mathbb{C}}$. There we may take ${\mathbb{C}}$ as a field without cohesion, which is enough to define complex differentiability. That is what I understand to be the point about Goursat theorem v.s. Cauchy’s original result. After this hard analysis is done, then lifting the definition of holomorphicity to complex manifolds could be done using the antiholomorphic differential.

Referring to Urs’ work on cohesion in complex geometry, the subleties of defining “holomorphic” as “complex-differentiability” could arise when studying functions between more general complex analytic space, as opposed to the traditional setting of complex manifolds.

]]>Edited biholomorphic function to follow the same format as diffeomorphism. In particular, this means that I qualified biholomorphic function to refer only to maps between complex manifolds. Is there a more general definition of holomorphic functions between complex analytic spaces?

]]>It is well known that a category can be defined as a certain simplicial set obtained by iterated fibred products which satisfies the internal horn filler condition; moreover, requiring horn filling for all horns (i.e., the Kan condition) one obtains the notion of groupoid. Then both the notions of category and groupoid should have an internalization in any category where one is able to arrange things in a way to have the required fibered products, and to state the horn filling condition.

This is what happens, e.g., when one defines Lie groupoids imposing that the source and target maps are submersions. Similarly one has a notion of Lie category, which by some reason seems to be less widely known of the more particular notion of Lie groupoid (maybe this is not surprising.. after all I suspect the notion of category is less known of that of group..). Another classical example are topological categories and groupoids.

Moving from categories and groupoids to oo-categories and oo-groupoids, one should have a similar internal simplicial object based notion of, e.g., Lie oo-groupoid. However, in the nLab the oo-sheaf point of view seems to be largely preferred to the internal Kan object point of view. Why is it so? is the oo-sheaf version just more general and powerful or there are problems with the internal version? I’m asking this since at internal infinity-groupoid it is said that a classical example of the internal Kan complex definition of oo-groupoid are Lie oo-groupoids, but then at Lie infinity-groupoid there is no trace of the internal Kan complex definition.

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