Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorEric
   • CommentTimeOct 25th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexI just googled >+descent manifold "transition function" which led me to the Wikipedia article on [connections](http://en.wikipedia.org/wiki/Connection_%28mathematics%29), where we find a reference: >The most abstract approach may be that suggested by Alexander Grothendieck, where a [Grothendieck connection](http://en.wikipedia.org/wiki/Grothendieck_connection) is seen as [descent](http://en.wikipedia.org/wiki/Descent_(category_theory)) data from infinitesimal neighbourhoods of the [diagonal](http://en.wikipedia.org/wiki/Diagonal); see ([Osserman 2004](http://en.wikipedia.org/wiki/Connection_(mathematics)#CITEREFOsserman2004)). From there, I could not resist having a look at [Osserman](http://www.math.ucdavis.edu/~osserman/math/connections.pdf). From there, I noticed a slight resemblance to some things [[ericforgy:differential envelope|I've worked on]]. _Warning: Entering shaky math zone. Help is appreciated.Apologies in advance for poor writing style, but I'm trying._ When your space admits a finite set of complete orthogonal projections $$1 = \sum_i \pi_i\quad\text{and}\quad \pi_i\circ\pi_j = \delta_{i j}\pi_i,$$ you can generate a free graded module of projections $\Omega$. On this graded module, you can define a universal "graph operator" given by the innocuous looking $$G = 1\otimes 1$$ which can be expanded into projections via $$G = \sum_{i,j} \pi_i\otimes\pi_j.$$ On certain special spaces (called "diamonds"), we can define a nilpotent map $d:\Omega^n\to\Omega^{n+1}$ via the graded commutator with the graph operator $$d\alpha \coloneqq [G,\alpha]$$ for any $\alpha\in\Omega$. Note, in particular, that for $f\in\Omega^0$, we have $$d f = [G,f] = 1\otimes f - f\otimes 1.$$ This is in the kernel of a multiplication map $m:\Omega^1\to\Omega^0$ given by $$m(f\otimes g) = f g.$$ Now, if we interpret $\pi_i\otimes\pi_j$ as a directed edge $\pi_i\to\pi_j$ (hence the name "graph operator" above), then the universal graph operator corresponds to a complete graph. The differential graded algebra corresponding to this universal graph operator is a universal differential envelope. Any other differential graded algebra of projections can be obtained from this one by setting some of the $\pi_i\otimes\pi_j$ to zero and taking appropriate quotients. I am tempted to associate the process of setting some directed edges to zero with the process of defining the neighborhood of the diagonal. In other words, setting an element of the diagonal to zero means it is not in the neighborhood. Does that make any sense? PS: Since $\Omega^0$ is generated by projections, we have $$f = \sum_i f_i \pi_i.$$ Therefore, $d f$ may be expressed in terms of projections via $$d f = 1\otimes f - f\otimes 1 = \sum_{i,j} (f_j - f_i) \pi_i \otimes \pi_j.$$

   I just googled

   +descent manifold “transition function”

   which led me to the Wikipedia article on connections, where we find a reference:

   The most abstract approach may be that suggested by Alexander Grothendieck, where a Grothendieck connection is seen as descent data from infinitesimal neighbourhoods of the diagonal; see (Osserman 2004).

   From there, I could not resist having a look at Osserman.

   From there, I noticed a slight resemblance to some things I’ve worked on.

   Warning: Entering shaky math zone. Help is appreciated.Apologies in advance for poor writing style, but I’m trying.

   When your space admits a finite set of complete orthogonal projections

   $$1 = \sum_i \pi_i\quad\text{and}\quad \pi_i\circ\pi_j = \delta_{i j}\pi_i,$$

   you can generate a free graded module of projections $\Omega$. On this graded module, you can define a universal “graph operator” given by the innocuous looking

   $$G = 1\otimes 1$$

   which can be expanded into projections via

   $$G = \sum_{i,j} \pi_i\otimes\pi_j.$$

   On certain special spaces (called “diamonds”), we can define a nilpotent map $d:\Omega^n\to\Omega^{n+1}$ via the graded commutator with the graph operator

   $$d\alpha \coloneqq [G,\alpha]$$

   for any $\alpha\in\Omega$. Note, in particular, that for $f\in\Omega^0$, we have

   $$d f = [G,f] = 1\otimes f - f\otimes 1.$$

   This is in the kernel of a multiplication map $m:\Omega^1\to\Omega^0$ given by

   $$m(f\otimes g) = f g.$$

   Now, if we interpret $\pi_i\otimes\pi_j$ as a directed edge $\pi_i\to\pi_j$ (hence the name “graph operator” above), then the universal graph operator corresponds to a complete graph. The differential graded algebra corresponding to this universal graph operator is a universal differential envelope.

   Any other differential graded algebra of projections can be obtained from this one by setting some of the $\pi_i\otimes\pi_j$ to zero and taking appropriate quotients.

   I am tempted to associate the process of setting some directed edges to zero with the process of defining the neighborhood of the diagonal. In other words, setting an element of the diagonal to zero means it is not in the neighborhood. Does that make any sense?

   PS: Since $\Omega^0$ is generated by projections, we have

   $$f = \sum_i f_i \pi_i.$$

   Therefore, $d f$ may be expressed in terms of projections via

   $$d f = 1\otimes f - f\otimes 1 = \sum_{i,j} (f_j - f_i) \pi_i \otimes \pi_j.$$
2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeOct 25th 2010
   • (edited Oct 25th 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexJust a quick comment (I did not carefully try to understand Eric's question yet). $n$lab also has a [[Grothendieck connection]] entry. The descent data in question are usually known as "costratification". Intuitive introduction is in the book by Berthelot and Ogus quoted there, paragraph 2.

   Just a quick comment (I did not carefully try to understand Eric’s question yet). $n$lab also has a Grothendieck connection entry. The descent data in question are usually known as “costratification”. Intuitive introduction is in the book by Berthelot and Ogus quoted there, paragraph 2.