http://www.physicsforums.com/showthread.php?t=168317

Hermann Weyl stated

“In these days the angel of toplogy and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.”

…

if you are a geometer, and have much experience with learning sheaf theory, and cohomology, you will understand what he is saying.

there are even geometric topoologists who dislike algebraic topology. I have tried to teach toric varieties to geometers and topologists who after seeing the definitions via spectra of various rings, asked, “OK, but where is the geometry? how do you get your HANDS on them?”

there is a feeling that algebraic methods take away intuition and render simple arguments too abstract. e.g. do you believe an irreducible non singular affine algebraic curve is really an integrally closed integral domain of krull dimension one?

or that the tangent bundle to a variety is really the set of k[e] valued points where k[e] = k[t](t^2) is the dual numbers? (actually this is fermat’s original definition, almost.)

or that a universal family of geometric objects should be regarded as a representable functor?

or that the right way to view a sheaf on a topological space is as a contravariant functor on category defined by the toopology where inclusions are the only morphisms?

You should, as this gives rise to the observation that one can generalize them to categories with more than one map between two objects, leading to the etale topology, and “stacks” where even single points have automorphisms.

these are needed to deal appropriately with local quotient spaces by groups acting with fixed points.

topologists tend to prefer homotopy to homology for this reaon, it is more geometric. Ed Brown Jr. considered his representation thoerem for cohomology as showing that cohomology was better than homology because being representable via homotopy showed that “it occurs in nature”.

“universal family” math

then one gets pages beginning:

[PDF]
arXiv:math.AG/0505084 v1 5 May 2005
File Format: PDF/Adobe Acrobat - View as HTML
math.AG/0505084. Transformation of algebraic Gromov-Witten invariants of …. with its universal family is the descent of the local standard models of …
arxiv.org/pdf/math/0505084 - Similar pages
[PDF]

arXiv:0710.3072v1 [math ...
File Format: PDF/Adobe Acrobat - View as HTML
arXiv:0710.3072v1 [math.AG] 16 Oct 2007 …. of G-equivariant coherent sheaves on M, with kernel the universal family Z ⊆ Hilb …
arxiv.org/pdf/0710.3072 - Similar pages
More results from arxiv.org »

Ars Mathematica
If you’re trying to look up the term universal family, my recommendation … I was looking at the stats for Ars Math, when I saw that we’ve had 525 posts. …
http://www.arsmathematica.net/ - Similar pages
Properties of sequence spaces in which l …
latter yields a weak wedge space universal family of spaces of the form Z for ….. Snyder, A.K.: Universal families for conull FK spaces. Trans. Am. Math. …
http://www.springerlink.com/index/U014371W676776X8.pdf - Similar pages

Wapedia - Wiki: Moduli space
Heuristically, if we have a space M for which each point m \in M … A fine moduli space is a space M which is the base of a universal family. …
wapedia.mobi/en/Moduli_space - 6k - Cached - Similar pages
Citebase - Harmonic Magnus Expansion on the Universal Family of …
Harmonic Magnus Expansion on the Universal Family of Riemann Surfaces … Based on record (harvested at): oai:arXiv.org:math/0603158 (2008-02-17) …
http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Amath%2F0603158 - 13k - Cached - Similar pages

Math 416, Fall 2004: Expanded Cumulative Syllabus
Math 416, Fall 2004: Expanded Cumulative Syllabus. Sept 8–10: … 2-universal hashing: A random hash function from a 2-universal family gives the correct …
http://www.eecs.umich.edu/~martinjs/math416/syl.html - 9k - Cached - Similar pages

p-adic variation of motives
Some of the most pressing open questions are: Can one establish the existence of the universal family (or any non-trivial family) of p-motives containing a …
http://www.birs.ca/workshops/2003/03w5104/ - 10k - Cached - Similar pages