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.
   • CommentAuthorUrs
   • CommentTimeJun 18th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexgave [[base change geometric morphism]] its own dedicated paragraph

   gave base change geometric morphism its own dedicated paragraph

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeJun 19th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI don't think that this addresses the full lack of generality at [[base change]], but that's OK for now.

   I don’t think that this addresses the full lack of generality at base change, but that’s OK for now.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 14th 2017
   • (edited Mar 14th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[base change]]_ the example ([here](https://ncatlab.org/nlab/show/base+change#AlongPointInclusionIntoBG)) that $$ \underset{\ast \to \mathbf{B}G}{\prod} \;\simeq\; [G,-]/G $$ In particular for $G = S^1$, then this is the [[cyclic loop space]] construction $$ \underset{\ast \to \mathbf{B}S^1}{\prod} \;\simeq\; \mathfrak{L}(-)/S^1 $$ I used to have this statement at _[[double dimensional reduction]]_, but since it's a special case of base change, it should be found there, too. This is elementary, and still connected to some rich constructions, as the relation to the cyclic loop space shows. I am wondering if this has been discussed anywhere else.

   I have added to base change the example (here) that

   $$\underset{\ast \to \mathbf{B}G}{\prod} \;\simeq\; [G,-]/G$$

   In particular for $G = S^1$, then this is the cyclic loop space construction

   $$\underset{\ast \to \mathbf{B}S^1}{\prod} \;\simeq\; \mathfrak{L}(-)/S^1$$

   I used to have this statement at double dimensional reduction, but since it’s a special case of base change, it should be found there, too.

   This is elementary, and still connected to some rich constructions, as the relation to the cyclic loop space shows. I am wondering if this has been discussed anywhere else.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 14th 2017
   • (edited Mar 14th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexCharles Rezk kindly points out that this kind of statement generalizes from base change along $\ast \to \mathbf{B}G$ to base change along $V/G \to \mathbf{B}G$ for every $G$-action on any $V$. I have added now statement and proof of this more general version [here](https://ncatlab.org/nlab/show/base+change#RightBaseChangeAlongUniversalFiberBundleProjection).

   Charles Rezk kindly points out that this kind of statement generalizes from base change along $\ast \to \mathbf{B}G$ to base change along $V/G \to \mathbf{B}G$ for every $G$-action on any $V$.

   I have added now statement and proof of this more general version here.