Talk:Categories of manifolds

Serious reservations

I have serious reservations about the suitability of an article on this topic. The present text consists mostly of original research by synthesis, which seems inevitable given the nature of the subject. Worse, trying to synthesize diverse phenomena, it does little more than introduce errors and confusion, rather than clarify List of manifolds#Categories_of_manifolds. Of paricular concern are statements dealing with symplectic manifolds.

The word category is applied both in its colloquial meaning, "collection of objects of like structure" and its more precise mathematical meaning. For the question of existense of a richer structure on a geometric object, for example, symplectic structure on a differentiable manifold or smooth structure on a topological manifold, functoriality is usually irrelevant. Hence talking about category in precise mathematical sense does not add anything substantial. It can, however, lead to errors.
I've never heard about, say, "category of exotic spheres" or "category of symmetric spaces", or even "category of homology manifolds": what are they doing here?
It is meaningful to discuss conditions under which, say, an orientable manifold has spin structure, or a topological manifold can be triangulated, but I think that this is best left to the individual articles dealing with these structures, as very little can said in general. Same remark applies to "α structure" vs "almost α structure" statements.
Several standard structures described in the article are not G-structures. For example, symplectic manifold is not a 2n-simensional differentiable manifold whose holonomy group reduces to Sp(2n): the key condition is that the symplectic form ω be closed.
It is far from clear what is meant by "the category of symplectic manifolds". This is not so much the deficiency of the article itself as the reflection on the current state of affairs (and perhaps on the shortcomings of the general category theory treatment). For example, from a certain point of view, the "correct" class of automorphisms of a symplectic manifold X is given by Hamiltonian self-maps, and from another, perhaps even more useful, by Lagrangian subvarieties of X × (−X). For Hom(X,Y), where X and Y are symplectic manifolds of different dimensions, the situation is worse.

Given these concerns, I propose that the author delete the article before anyone else edits it — this is relatively uncomplicated. Any new material can be merged to the corresponding specialized articles. Arcfrk (talk) 00:31, 22 November 2007 (UTC)[reply]

The article admittedly read more like an essay than an encyclopedia entry; I’ve accordingly moved it to wikibooks:Topology/Manifolds/Categories of Manifolds and made this entry a redirect to List of manifolds#Categories of manifolds.

The content of this article is important, as it explains how Geometric Topologists organize the various classes of manifolds and structures they see, and category theory is fundamental in how one approaches this. This merits a unified discussion somewhere, rather than only scattered notes – there is value in specialized articles, and value in more general treatments – Wikibooks is likely best-placed for this latter.

Your point that other notions of manifold don’t fit so well into the categorical structure is well-taken; I include them because there do naturally fit into a progression of structures (as in G-structures).

—Nils von Barth (nbarth) (talk) 02:16, 7 December 2009 (UTC)[reply]