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1. • CommentRowNumber1.
   • CommentAuthorStephan A Spahn
   • CommentTimeApr 3rd 2012
   • (edited Apr 3rd 2012)
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItexI looked at the [[homotopy groups in an (infinity,1)-topos]] and have the questions how the notions defined there relate to each other. > We discuss below both cases. These are [[categorical homotopy groups in an (∞,1)-topos]] and [[geometric homotopy groups in an (∞,1)-topos]] with the hint that the geometric homotopy generalizes to [[étale homotopy]] which in turn is an instance of [[shape theory]]. So do we want to look at the latter at some generalized homotopy theory? > The case of categorical homotopy groups is fully understood, for the case of geometric homotopy groups at the moment only a few aspects are in the literature, more is in the making. Some authors of this page ([[Urs Schreiber|U.S.]]) thank [[Richard Williamson]] for pointing this out. Maybe it has become more clear since this entry has been written where these notions coincide and where they differ?

   I looked at the homotopy groups in an (infinity,1)-topos and have the questions how the notions defined there relate to each other.

   We discuss below both cases.

   These are categorical homotopy groups in an (∞,1)-topos and geometric homotopy groups in an (∞,1)-topos with the hint that the geometric homotopy generalizes to étale homotopy which in turn is an instance of shape theory. So do we want to look at the latter at some generalized homotopy theory?

   The case of categorical homotopy groups is fully understood, for the case of geometric homotopy groups at the moment only a few aspects are in the literature, more is in the making. Some authors of this page (U.S.) thank Richard Williamson for pointing this out.

   Maybe it has become more clear since this entry has been written where these notions coincide and where they differ?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 3rd 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi Stephan, in principle there is no mystery here, maybe let me know in more detail what you are after. In the special case where we are looking at a [[locally infinity-connected (infinity,1)-topos]] $\mathbf{H}$ the situation is as simple as this: we have that $\infty$-functor $\Pi : \mathbf{H} \to \infty Grpd$. And the geometric homotopy in $\mathbf{H}$ is simply the categorical homotopy in $\infty Grpd$ under this functor. If the topos is not locally $\infty$-connected, there is a substitute for that $\Pi$ which is a tad more involved, but mainly behaves as $\Pi$ does. This is what is described at [[shape of an (infinity,1)-topos]]. So, that's how the two notions are related. Concerning your question on whether we regard this as a "generalized homotopy theory": in principle every $(\infty,1)$-category is a "generalized homotopy theory", in that it generalizes the standard homotopy theory which takes place in $\infty Grpd \simeq Top$. So, if you want, one way to look at the "geometric homotopy theory" is as a way to take a generalized homotopy theory (say that given by $\mathbf{H}$ above) and then map it to "ordinary homotopy theory" in some way. That's what $\Pi$ / shape does. Closely related to this is the fact that we may think of $\Pi : \mathbf{H} \to \infty Grpd \stackrel{\simeq}{\to} Top$ as being _geometric realization_. It takes an object in a "generalized homotopy theory" and forms the homotopy type of a topological space (object in "ordinary homotopy theory") that "best approximates it" in some precise sense. Does that help? I or others here can say more if you give more details on what you are after.

   Hi Stephan,

   in principle there is no mystery here, maybe let me know in more detail what you are after.

   In the special case where we are looking at a locally infinity-connected (infinity,1)-topos $\mathbf{H}$ the situation is as simple as this:

   we have that $\infty$-functor $\Pi : \mathbf{H} \to \infty Grpd$. And the geometric homotopy in $\mathbf{H}$ is simply the categorical homotopy in $\infty Grpd$ under this functor.

   If the topos is not locally $\infty$-connected, there is a substitute for that $\Pi$ which is a tad more involved, but mainly behaves as $\Pi$ does. This is what is described at shape of an (infinity,1)-topos.

   So, that’s how the two notions are related.

   Concerning your question on whether we regard this as a “generalized homotopy theory”: in principle every $(\infty,1)$-category is a “generalized homotopy theory”, in that it generalizes the standard homotopy theory which takes place in $\infty Grpd \simeq Top$. So, if you want, one way to look at the “geometric homotopy theory” is as a way to take a generalized homotopy theory (say that given by $\mathbf{H}$ above) and then map it to “ordinary homotopy theory” in some way. That’s what $\Pi$ / shape does. Closely related to this is the fact that we may think of $\Pi : \mathbf{H} \to \infty Grpd \stackrel{\simeq}{\to} Top$ as being geometric realization. It takes an object in a “generalized homotopy theory” and forms the homotopy type of a topological space (object in “ordinary homotopy theory”) that “best approximates it” in some precise sense.

   Does that help? I or others here can say more if you give more details on what you are after.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 3rd 2012
   • (edited Apr 3rd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOne more comment, which might be useful, or might not be useful: in some situations we may think of geometric homotopy as being categorical homotopy "after $\mathbb{A}^1$-localization" in the sense of [[A1-homotopy theory]]. This is the case whenever the "geometric homotopies" are given by actual geoemtric paths incarnated in terms of a line object $\mathbb{A}^1$. For instance in $\mathbf{H} = $ [[Smooth∞Grpd]] = $Sh_{\infty}(SmthMfd)$ the ordinary real line serves as a line object $$ \mathbb{A}^1 := \mathbb{R}^1 $$ and the [[disctete ∞-groupoid]]-inclusion $$ \infty Grpd \simeq Sh_\infty(SmthMfd)_{\mathbb{R}^1} \stackrel{\overset{\Pi}{\leftarrow}}{\underset{Disc}{\hookrightarrow}} Sh_\infty(SmthMfd) \simeq Smooth \infty Grpd $$ is precisely the [[A1-homotopy theory]] localization with respect to this differential geometric line object. So "[[motivic cohomology]]" in the context of diferential geometry is just ordinary cohomology theory. Conversely, if you have another $\infty$-topos which is not locally $\infty$-connected but does come equipped with a line object $\mathbb{A}^1$ (such as the one over the [[Nisnevich site]], which is the context of ordinary $\mathbb{A}^1$-homotopy theory), you may regard its $\mathbb{A}^1$-localization as producing some kind of "generalized geometric homotopy theory" relative to that $\mathbb{A}^1$.

   One more comment, which might be useful, or might not be useful:

   in some situations we may think of geometric homotopy as being categorical homotopy “after $\mathbb{A}^1$-localization” in the sense of A1-homotopy theory.

   This is the case whenever the “geometric homotopies” are given by actual geoemtric paths incarnated in terms of a line object $\mathbb{A}^1$.

   For instance in $\mathbf{H} =$ Smooth∞Grpd = $Sh_{\infty}(SmthMfd)$ the ordinary real line serves as a line object

   $$\mathbb{A}^1 := \mathbb{R}^1$$

   and the disctete ∞-groupoid-inclusion

   $$\infty Grpd \simeq Sh_\infty(SmthMfd)_{\mathbb{R}^1} \stackrel{\overset{\Pi}{\leftarrow}}{\underset{Disc}{\hookrightarrow}} Sh_\infty(SmthMfd) \simeq Smooth \infty Grpd$$

   is precisely the A1-homotopy theory localization with respect to this differential geometric line object.

   So “motivic cohomology” in the context of diferential geometry is just ordinary cohomology theory. Conversely, if you have another $\infty$-topos which is not locally $\infty$-connected but does come equipped with a line object $\mathbb{A}^1$ (such as the one over the Nisnevich site, which is the context of ordinary $\mathbb{A}^1$-homotopy theory), you may regard its $\mathbb{A}^1$-localization as producing some kind of “generalized geometric homotopy theory” relative to that $\mathbb{A}^1$.

4. • CommentRowNumber4.
   • CommentAuthorStephan A Spahn
   • CommentTimeApr 5th 2012
   • (edited Apr 5th 2012)
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItexHi Urs, sorry for the slow reply, I was in [a model of type theory in simplicial sets](http://ncatlab.org/nlab/show/T.+Streicher+-+a+model+of+type+theory+in+simplicial+sets+-+a+brief+introduction+to+Voevodsky%27+s+homotopy+type+theory) as preparation for the upcoming mini-school in Swansea. > let me know in more detail what you are after. My idea was to see if there is some notion of ''formally étale homotopy'' making sense. In light of > $\Pi : \mathbf{H} \to \infty Grpd$. And the geometric homotopy in $\mathbf{H}$ is simply the categorical homotopy in $\infty Grpd$ under this functor. > So, if you want, one way to look at the "geometric homotopy theory" is as a way to take a generalized homotopy theory (say that given by $\mathbf{H}$ above) and then map it to "ordinary homotopy theory" in some way. a candidate for a functor $\Pi$ defined on the infinitesimal neighbourhood $H_th$ of $H$ would be $\Pi_inf$ since we have a factorization of $\Pi_th$ into $H_th\xrightarrow{\Pi_inf}H\xrightarrow{Pi}\infty Grpd$. Then one could consider what this factorization does with the constructions from [...in terms of monodromy and Galois theory](http://ncatlab.org/nlab/show/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos#Monodromy). But this was just a vague idea and I do not follow it for the moment. The larger context of this has been that I was looking for an exit out of the ''dead end'' our considerations of the formally étale morphisms in a cohesive ($\infty$,1)-topos led us into. For those who maybe are listening in: The problem was to see if or if not the sub-(∞,1)-category $fEt/X$ of the over-(∞,1)-topos $H/X$ is an (∞,1)-topos.

   Hi Urs,

   sorry for the slow reply, I was in a model of type theory in simplicial sets as preparation for the upcoming mini-school in Swansea.

   let me know in more detail what you are after.

   My idea was to see if there is some notion of ”formally étale homotopy” making sense. In light of

   $\Pi : \mathbf{H} \to \infty Grpd$. And the geometric homotopy in $\mathbf{H}$ is simply the categorical homotopy in $\infty Grpd$ under this functor. So, if you want, one way to look at the “geometric homotopy theory” is as a way to take a generalized homotopy theory (say that given by $\mathbf{H}$ above) and then map it to “ordinary homotopy theory” in some way.

   a candidate for a functor $\Pi$ defined on the infinitesimal neighbourhood $H_th$ of $H$ would be $\Pi_inf$ since we have a factorization of $\Pi_th$ into $H_th\xrightarrow{\Pi_inf}H\xrightarrow{Pi}\infty Grpd$. Then one could consider what this factorization does with the constructions from …in terms of monodromy and Galois theory. But this was just a vague idea and I do not follow it for the moment.

   The larger context of this has been that I was looking for an exit out of the ”dead end” our considerations of the formally étale morphisms in a cohesive ($\infty$,1)-topos led us into. For those who maybe are listening in: The problem was to see if or if not the sub-(∞,1)-category $fEt/X$ of the over-(∞,1)-topos $H/X$ is an (∞,1)-topos.