# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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 $\mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type$ 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.

• 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 $G Act_\infty \simeq \infty Grpd_{/ B G}$.

(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 $G$-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.

• 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 $H$ 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.)

• 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.