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March 2, 2010

Rational Homotopy Theory in an (oo,1)-Topos

Posted by Urs Schreiber

In the previous entry John described higher gauge theory with the declared intent of not emphasizing its higher category theory.

The other extreme of this has its own charms: try to describe higher gauge theory entirely using formal category theory on an ambient \infty-topos. I have been entertaining myself with searching for this intrinsic \infty-topos-theoretic formulation. The present state of my understanding is summarized on this page:

This lists structures that are available on purely formal grounds in an (,1)(\infty,1)-topos: its shape, its cohomology, its homotopy, its rational homotopy, its differential cohomology.

Here differential cohomology in an (,1)(\infty,1)-topos is just another word for higher gauge theory .

In the last days maybe I was able to fill what used to be a gap in the abstract story that I was trying to tell. It helped to read the remarkable recent article

and its emphasis of the issue discussed in the remarkable

This provides a nice point of view on rational homotopy theory from \infty-topos theory. Some aspects of this I had understood before. For instance the left adjoint \infty-functor that produces the global function dg-algebra on an \infty-stack discussed there is what I used to call the Chevalley-Eilenberg algebra of an \infty-stack. But after reading this again now I obtained a clear simple abstract picture that I did not quite have before.

On the page

there is first a section that describes this abstract picture. Then there is a section that recalls key definitions and results from Bertrand Toën’s article from this perspective.

Posted at March 2, 2010 12:26 PM UTC

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Re: Rational Homotopy Theory in an (oo,1)-Topos

Where is the rational? How does a reflexive embedding give us a characteristic 0 field? or is this just the usual overuse of a single word?
Alternatively, why isn’t this just
Homotopy theory in an (oo,1)-topos?

Posted by: jim stasheff on March 4, 2010 2:54 PM | Permalink | Reply to this

Re: Rational Homotopy Theory in an (oo,1)-Topos

Where is the rational?

There is of course the choice of ground field kk in L:=(Alg k Δ) op \mathbf{L} := (Alg_k^\Delta)^{op}.

The relation to genuine rational homotopy as in Toën’s theorem 2.5.1 is obtained with k=k = \mathbb{Q}.

But other choices of kk give analogous interesting structure. For instance for kk of finite characteristic and algebraically closed, one still gets that the unit XΓSpec𝒪LConstXX \to \Gamma Spec \mathcal{O} LConst X of the total adjunction

LSpec𝒪HΓLConstGrpdTop cg \mathbf{L} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \simeq Top_{cg}

is an equivalence on kk-cohomology groups, hence is “rationalization” with respect to kk.

Alternatively, why isn’t this just Homotopy theory in an (oo,1)-topos?

Right, for the above reason.

The full homotopy theory of H\mathbf{H} is instead encoded differently, as described at homotopy groups in an (,1)(\infty,1)-topos.

One way to say it is to assume that the terminal geometric morphism HΓLConstGrpd\mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd is essential meaning that there is a further left adjoint Π\Pi to LConstLConst

HΓLConstΠGrpd. \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \,.

Then for any XHX \in \mathbf{H} we have its geometric homotopy groups encoded in the fundamental \infty-groupoid Π(X)\Pi(X) and |Π(X)|Top cg|\Pi(X)| \in Top_{cg} is the geometric realization of the object in H\mathbf{H}.

So then the rational homotopy of XHX \in \mathbf{H} is computed by chasing XX along the full S-shaped path of the total diagram

LSpec𝒪HΓLConstΠGrpdTop cg. \mathbf{L} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \simeq Top_{cg} \,.

How does a reflexive embedding give us a characteristic 0 field?

Well, so far the reflective embedding is built after a kk has been chosen by hand.

But I think that one should be able to go the other way round, as you seem to indicate, and determine the kk from an abstract property of H\mathbf{H}. It should be determined by the line object that is encoded by Π\Pi. I can say more about this a little later.

Posted by: Urs Schreiber on March 4, 2010 5:59 PM | Permalink | Reply to this

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