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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2012

    started an entry infinity-action

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2012
    • (edited Sep 16th 2012)

    I have been expanding at infinity-action in an attempt to make the discussion more comprehensive and more coherent. But not done yet.

    For instance I made explicit that for c:BGV(c):Type\mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type an \infty-action we have

    • the dependent product

      c:BGV(c):Type \vdash \prod_{\mathbf{c} : \mathbf{B}G}V(\mathbf{c}) : Type

      is the type of invariants of the action;

    • the dependent sum

      c:BGV(c):Type \vdash \sum_{\mathbf{c} : \mathbf{B}G}V(\mathbf{c}) : Type

      is the quotient of the action

    And hence for V 1V_1 and V 2V_2 two actions we have that

    • c:BGV 1(c)V 2(c):Type\vdash \prod_{\mathbf{c} : \mathbf{B}G} V_1(\mathbf{c}) \to V_2(\mathbf{c}) : Type

      is the type of GG-homomorphisms (of GG-equivariant maps);

    • c:BGV 1(c)V 2(c):Type\vdash \sum_{\mathbf{c} : \mathbf{B}G} V_1(\mathbf{c}) \to V_2(\mathbf{c}) : Type

      is the quotient of all maps by the GG-cojugation action.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2012

    Added at infinity-action brief remarks in new subsections

    This is more to remind myself to come back to it later. No time right now.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 5th 2014

    added to infinity-action under a new section References – For discrete geometry pointers to articles that discuss GAct Grpd /BGG Act_\infty \simeq \infty Grpd_{/ B G}.

    (copied the same also to the end of the citations at principal infinity-bundle).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 11th 2014
    • (edited Mar 11th 2014)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2014
    • (edited Apr 14th 2014)

    added now also a new section References – For actions of topological groups with the following text (which might also go into a Properties-section, admittedly):


    [ text extracted from entry ]

    That GG-actions for GG a topological group in the sense of G-spaces in equivariant homotopy theory (and hence with GG not regarded as the geometrically discrete ∞-group of its underying homotopy type ) are equivalently objects in the slice (∞,1)-topos over BG\mathbf{B}G is Elmendorf’s theorem together with the fact, highlighted in this context in

    that

    GSpacePSh (Orb G)PSh (Orb /BG)PSh (Orb) /BG G Space \simeq PSh_\infty(Orb_G) \simeq PSh_\infty(Orb_{/\mathbf{B}G}) \simeq PSh_\infty(Orb)_{/\mathbf{B}G}

    is therefore the slice of the \infty-topos over the slice of the global orbit category over BG\mathbf{B}G.

    !include equivariant homotopy theory – table

    See at equivariant homotopy theory for more references along these lines.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2014
    • (edited Nov 4th 2014)

    added a little observation – here – on how the automorphism action on an object in a slice is given by descending the corresponding map to the homotopy quotient of the induced action on the dependent sum.

    (This innocent statement gives symplectic reduction for prequantum n-bundles)

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 4th 2015

    There is a nice formalization of linearization of \infty-actions in axiomatics of differential cohesion. I have added it to the entry here.

    (The text deserves further polishing, but I am in a haste now as my battery is dying.)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 4th 2015

    I have now written this (linearization of \infty-actions) out in more detail in section 3.10.5 of dcct (pdf).

    In section 3.10.11 I use this to axiomatize Cartan geometry in differential cohesion, via the key example (3.10.45) of the canonical linearized HH action on 𝔤/𝔥\mathfrak{g}/\mathfrak{h} .

    (I have been fiddling with the axiomatization of Cartan geometry in homotopy-type theory a bit, as you may recall. My first versions were not so good, but I think now I am converging.)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2015
    • (edited Apr 16th 2015)

    Added an expositional section Examples – Discrete group actions on sets which spells out in detail how the general abstract slicing perspective recovers ordinary permutation representations.

    I should maybe copy this also to, or at least link to also from, action groupoid and permutation representation.

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