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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics 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.
   • CommentAuthorAndrew Stacey
   • CommentTimeJun 21st 2011
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexI got a question by email about the _equivariant_ tubular neighbourhoods in loop spaces (as opposed to those defined using [[propagating flows]] so I figured it was time to nLabify that section of [[differential topology of mapping spaces]]. Of course, in so doing I figured out a generalisation: given a fibre bundle $E \to B$, everything compact, we consider smooth maps $E \to M$ which are constant on fibres. This is a submanifold of the space of all smooth maps $E \to M$. Assuming we can put a suitable measure on the fibres of $E$, then we can define a tubular neighbourhood of this submanifold. Details at [[equivariant tubular neighbourhoods]]. Title may be a bit off now, but it's that because the original case was for the fibre bundle $S^1 \to S^1$ with fibre $\mathbb{Z}_n$. This entry is also notable because I produced it using a whole new LaTeX-to-iTeX converter. Details on the relevant thread.

   I got a question by email about the equivariant tubular neighbourhoods in loop spaces (as opposed to those defined using propagating flows so I figured it was time to nLabify that section of differential topology of mapping spaces. Of course, in so doing I figured out a generalisation: given a fibre bundle $E \to B$, everything compact, we consider smooth maps $E \to M$ which are constant on fibres. This is a submanifold of the space of all smooth maps $E \to M$. Assuming we can put a suitable measure on the fibres of $E$, then we can define a tubular neighbourhood of this submanifold.

   Details at equivariant tubular neighbourhoods. Title may be a bit off now, but it’s that because the original case was for the fibre bundle $S^1 \to S^1$ with fibre $\mathbb{Z}_n$.

   This entry is also notable because I produced it using a whole new LaTeX-to-iTeX converter. Details on the relevant thread.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 22nd 2011
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   Author: DavidRoberts
   Format: MarkdownItexThe displayed equation above lemma 1 (in version 1) is a bit wonky on the spacing, but apart from that a quick scan showed it to be very nice. Well done!

   The displayed equation above lemma 1 (in version 1) is a bit wonky on the spacing, but apart from that a quick scan showed it to be very nice. Well done!

3. • CommentRowNumber3.
   • CommentAuthorAndrew Stacey
   • CommentTimeJun 22nd 2011
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   Author: Andrew Stacey
   Format: MarkdownItexAagh. I keep forgetting that `\text` behaves ever so slightly differently with regard to swallowing spaces. Fixed. Also added a picture to try to illustrate some of the maps.

   Aagh. I keep forgetting that \text behaves ever so slightly differently with regard to swallowing spaces. Fixed.

   Also added a picture to try to illustrate some of the maps.