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started an entry infinity-action
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 $\mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type$ an $\infty$-action we have
the dependent product
$\vdash \prod_{\mathbf{c} : \mathbf{B}G}V(\mathbf{c}) : Type$
is the type of invariants of the action;
the dependent sum
$\vdash \sum_{\mathbf{c} : \mathbf{B}G}V(\mathbf{c}) : Type$
is the quotient of the action
And hence for $V_1$ and $V_2$ two actions we have that
$\vdash \prod_{\mathbf{c} : \mathbf{B}G} V_1(\mathbf{c}) \to V_2(\mathbf{c}) : Type$
is the type of $G$-homomorphisms (of $G$-equivariant maps);
$\vdash \sum_{\mathbf{c} : \mathbf{B}G} V_1(\mathbf{c}) \to V_2(\mathbf{c}) : Type$
is the quotient of all maps by the $G$-cojugation action.
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.
added to infinity-action under a new section References – For discrete geometry pointers to articles that discuss $G Act_\infty \simeq \infty Grpd_{/ B G}$.
(copied the same also to the end of the citations at principal infinity-bundle).
…
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 $G$-actions for $G$ a topological group in the sense of G-spaces in equivariant homotopy theory (and hence with $G$ not regarded as the geometrically discrete ∞-group of its underying homotopy type ) are equivalently objects in the slice (∞,1)-topos over $\mathbf{B}G$ is Elmendorf’s theorem together with the fact, highlighted in this context in
that
$G Space \simeq PSh_\infty(Orb_G) \simeq PSh_\infty(Orb_{/\mathbf{B}G}) \simeq PSh_\infty(Orb)_{/\mathbf{B}G}$is therefore the slice of the $\infty$-topos over the slice of the global orbit category over $\mathbf{B}G$.
!include equivariant homotopy theory – table
See at equivariant homotopy theory for more references along these lines.
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)
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.)
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 $H$ action on $\mathfrak{g}/\mathfrak{h}$ .
(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.)
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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