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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
   • CommentTimeAug 23rd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexPage created, but author did not leave any comments. <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory">current</a>

   Page created, but author did not leave any comments.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 23rd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have expanded the statement of the theorem, including also the statement that the derived adjunction unit is rationalization (on connected nilpotent rationally finite homotopy types) <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/2">v2</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory">current</a>

   I have expanded the statement of the theorem, including also the statement that the derived adjunction unit is rationalization (on connected nilpotent rationally finite homotopy types)

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 7th 2021
   • (edited Jul 7th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added ([here](https://ncatlab.org/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory#ChangeOfScalars)) a revisionistic statement of Bousfield-Gugenheim's Lemma 11.7, in a new section "Change of scalars". <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/7">v7</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory">current</a>

   I have added (here) a revisionistic statement of Bousfield-Gugenheim’s Lemma 11.7, in a new section “Change of scalars”.

   diff, v7, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 8th 2021
   • (edited Jul 8th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added ([here](https://ncatlab.org/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory#kChernDoldCharacterFactorsThroughRationalChernDoldCharacter)) a re-casting of the Bousfield-Gugenheim extension-of-scalars-Lemma as the statement that the real [[Chern-Dold character]] in non-abelian cohomology factors through the rational Chern-Dold character via Deligne-Griffiths-Morgan-Sullivan's "tensoring a homotopy type with $\mathbb{R}$"-operation. <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/8">v8</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory">current</a>

   I have added (here) a re-casting of the Bousfield-Gugenheim extension-of-scalars-Lemma as the statement that the real Chern-Dold character in non-abelian cohomology factors through the rational Chern-Dold character via Deligne-Griffiths-Morgan-Sullivan’s “tensoring a homotopy type with $\mathbb{R}$“-operation.

   diff, v8, current

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 8th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexDefinition 2.1 says > nilpotent: the fundamental group π1(X) is a nilpotent group; Don't we also need the action on higher homotopy groups to be nilpotent?

   Definition 2.1 says

   nilpotent: the fundamental group π1(X) is a nilpotent group;

   Don’t we also need the action on higher homotopy groups to be nilpotent?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 8th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, have added the missing clause now. <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/9">v9</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory">current</a>

   Yes, have added the missing clause now.

   diff, v9, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 14th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded pointer to: * [[Jacob Lurie]], Prop. 1.3.3, Cor. 1.3.4 in: DAG XIII: _[[Rational and p-Adic Homotopy Theory]]_, 2011 ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XIII.pdf), [[Lurie_RationalHomotopyTheory.pdf:file]]) <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/11">v11</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory">current</a>

   Added pointer to:

   • Jacob Lurie, Prop. 1.3.3, Cor. 1.3.4 in: DAG XIII: Rational and p-Adic Homotopy Theory, 2011 (pdf, Lurie_RationalHomotopyTheory.pdf:file)

   diff, v11, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 26th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Edgar Brown]], [[Robert H. Szczarba]], _Real and Rational Homotopy Theory_, Chapter 17 in: *[[Handbook of Algebraic Topology]]*, Elsevier, (1995) 869-915 ([doi:10.1016/B978-044481779-2/50018-3](https://doi.org/10.1016/B978-044481779-2/50018-3), [ZB](http://mirrors.library.cornell.edu/ZMATH/zmath/en/zmath/search/?an=0865.55009)) <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory/13">v13</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory">current</a>

   added pointer to:

   • Edgar Brown, Robert H. Szczarba, Real and Rational Homotopy Theory, Chapter 17 in: Handbook of Algebraic Topology, Elsevier, (1995) 869-915 (doi:10.1016/B978-044481779-2/50018-3, ZB)

   diff, v13, current