I added a link to this answer for how to construct the action coherently.

]]>Sure, I added a parenthetical.

]]>Is that just me, or might the reader of

…inducing equivalences $Map(Y,X)^K \simeq Map(Y,X^K)$ for all $Y\in C$…

be given a little help with the left term? At first I was looking about for an object of $C$ to form the power of.

]]>Ok, I added some discussion of this perspective to the “Definition” section, but I didn’t try to rewrite the “as a mapping space object” section, so there is a bit of dissonance.

]]>Thanks. Yes, this entry could do with some polishing and re-organization.

]]>I added to free loop space object an intuitive discussion of how $\mathcal{L}X$ becomes a group object in $C/X$ and acts on all other objects of $C/X$. Can anyone give a reference where this is worked out precisely and coherently?

When I have some time, I would like to reorganize free loop space object a bit: I think the description of at “as a mapping space object” deserves to be up in the Definition section. In particular it’s not specific to $\infty$-toposes, but makes sense as a power in any $(\infty,1)$-category. It also yields another definition in terms of ordinary conical limits, namely as the equalizer of the identity map of $X$ with itself.

]]>before I forget: I had added a brief remark on the formulation of free loop spaces in Homotopy type theory

]]>Where does the ’$S^1$’ come from to give you $LConst S^1$?

This is discussed in detail in the entry, starting here.

Under $Top \simeq \infty Grpd$ the $S^1$ is just the $S^1$! So in $\infty Grpd$ you may want to think of it as the fundamental $\infty$-groupoid $\Pi(S^1_{Top}) \simeq \mathbf{B}\mathbb{Z}$ of the topological circle.

But what makes all derived loop space and Hochschild cohomology yoga tick is the fact that

$S^1 \simeq * \coprod_{* \coprod *} *$as an $\infty$-pushout. One way to see this formally is to model the $\infty$-pushout as a homotopy pushout and model that by an ordinary pushout of a resolved diagram in $sSet$. That resolved diagram can be taken to be

$\array{ * \coprod * &\to& * \\ \downarrow && \downarrow \\ \Delta[1] &\to& \Delta[1]/\partial \Delta[1] }$and so this way appears the model for $S^1$ that you have in mind.

]]>Where does the ’$S^1$’ come from to give you $LConst S^1$? Is it the the obvious quotient of an interval with distinct endpoints?

]]>at free loop space object I formalized the discussion of how in an $(\infty,1)$-topos the free space object $\mathcal{L}X$ is given by the internal hom object $[LConst S^1 , X]$.

This is now in the section Free loop space object – In an $(\infty,1)$-topos.

(The discussion of the induced $S^1$-action currently given there should be improved by discussing that $LConst S^1$ has a canonical cogroup object structure.)

]]>But I also suggest reading together with it those crystals and D-modules notes to which I posted links to in nlab few days ago (e.g. to D-module) and left a note in nforum. The point is that there is not only linear version (D-modules and crystals of qcoh modules), but also nonlinear (D-schemes and crystals of schemes).

http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19%28Crystals%29.pdf

]]>I read it on Monday evening till 3:30 am and I am tired now.

]]>I am busy reading

David Ben-Zvi, David Nadler: Loop Spaces and Connections.

Lots of answers in there...

]]>Domenico,

yes, I agree, that's a way to put it.

But maybe we can start out saying that a priori all of acts on , while the "circle action" proper is only the part . Because the other parts of will have a role to play, too. As we said, the -grading on is induced from the action of what on the homology of the circle acts by , while the differential is that induced by . The first, even, Lie algebra element should correspond to nontrivial objects in .

]]>with this restriction (which we could denote to use a notation inspired by the one which is customary in differential topology, where one usually writes ), the only object of is the identity functor and one has . moreover, having only , the categorical circle action on is , which also fits with having be -invariant.

waiting for feedback here before editing the page according to the above. ]]>

okay, fixed.

So I removed your first query box, David. If you disagree, please put it back in.

Also a quick reply in your second query box.

]]>Also I added a bit about AUT(Pi_1(S^1)) after the 'exercise:...' statement.

David, thanks, of course I am being stupid and for we just have a group homomorphism, not an automorphism.

It's not a drama, we could also be talking about endomorphisms. But I think I'll just fix this the way you indicate...

]]>prove is equivariant with respect to the categorical circle action (this should be almost immediate)

I almost missed this, sorry: yes, this statement will be almost tautology, no matter what precisely the categorical circle action actually is concretely.

Because the map is just post-composition with :

If you wish, in the end it is the exchange law that says that it doesn't matter whether you first have a transformation and then postcompose or first postcompose and then have the transformation.

]]>There is a neat model of Drinfel'd of cyclic objects where a usual circle comes via a categorical limit construction from finite approximations by discrete sets of n points around the circle; a paper few years ago on the arXiv. In your very interesting reasoning above I suspect you are talking also about inertia groupoid of the original groupoid as a model for loop groupoid of original groupoid. I did not hear word inertia groupoid (twisted sectors in the Harvey, Vafa etc. physics parlance) though (but is a good advertising for connections to physics).

]]>I have now fixed some typos -- in case you were wondering :-)

So in words maybe it's good to say it this way:

if we use the skeleton as a model for what could also be modeled as then:

the objects of -- labeled by some -- describe maps from the circle onto itself that fix the basepoint and have winding number

the morphisms in -- labeled by some -- describe a rigid rotation of the loop -times around the circle.

It is due to being a skeleton, that only these integer-labeled maps appear, of course. But maybe it's actually useful to make this model explicit.

We should in parallel make the situation for the equivalent model explicit. That will produce a situation that looks much more like the naive circle action that one expects. But together probably both models illustrate nicely how we have to distinguish here between automorphisms of the circle (maps with winding number) and genuine rotations of the circle.

]]>