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nLab > Latest Changes: fundamental (oo,1)-category

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeSep 30th 2010
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Author: Urs
Format: MarkdownItexcreated [[fundamental (infinity,1)-category]] This is supposed to propose the evident definition. But have a critical look.

created fundamental (infinity,1)-category

This is supposed to propose the evident definition. But have a critical look.
- CommentRowNumber2.
- CommentAuthorMike Shulman
- CommentTimeSep 30th 2010
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Author: Mike Shulman
Format: MarkdownItexMaybe I am not understanding the definition, but it does not seem to me to define a simplicial set. It looks like you are making $\Delta^{k}_{top}$ directed in such a way that the only directed paths are those which land inside the spine. But the cosimplicial coface maps do not necessarily take spines into spines, so I don't think this gives you a cosimplicial object in your category $DTop$. E.g. a 2-simplex in $\Pi(X,D)$ would be a 2-simplex in $X$ whose 0- and 2-faces are directed paths, but not the 1-face.

Maybe I am not understanding the definition, but it does not seem to me to define a simplicial set. It looks like you are making $\Delta^{k}_{top}$ directed in such a way that the only directed paths are those which land inside the spine. But the cosimplicial coface maps do not necessarily take spines into spines, so I don’t think this gives you a cosimplicial object in your category $DTop$ . E.g. a 2-simplex in $\Pi(X,D)$ would be a 2-simplex in $X$ whose 0- and 2-faces are directed paths, but not the 1-face.
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeSep 30th 2010
- (edited Sep 30th 2010)
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Author: Urs
Format: MarkdownItexRight, I should say that the directed paths in $\Delta^k_{top}$ are those running along any of the 1-faces in the direction of increasing vertex number.

Right, I should say that the directed paths in $\Delta^k_{top}$ are those running along any of the 1-faces in the direction of increasing vertex number.
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeSep 30th 2010
- (edited Sep 30th 2010)
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Author: Urs
Format: MarkdownItexSo: Directed paths in $\Delta^k_{Top}$ are those that factor order-preservingly through its 1-skeleton.

So:

Directed paths in $\Delta^k_{Top}$ are those that factor order-preservingly through its 1-skeleton.
- CommentRowNumber5.
- CommentAuthorMike Shulman
- CommentTimeSep 30th 2010
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Author: Mike Shulman
Format: MarkdownItexOkay. Now I agree that it's a quasicategory, but it seems a little odd to me to allow homotopies between directed paths that pass through undirected paths along the way.

Okay. Now I agree that it’s a quasicategory, but it seems a little odd to me to allow homotopies between directed paths that pass through undirected paths along the way.
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeSep 30th 2010
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Author: Urs
Format: MarkdownItex> but it seems a little odd to me to allow homotopies between directed paths that pass through undirected paths along the way. Yes, I was wondering about that, too. First my intuition also was that homotopies should be directed. But then the fact that this definition is so evident (well, ahem, I misstated it first, but that's my fault, not the definition's) made me rethink. Also, given the discussion with Tim in the other thread, it seems that also requiring the homotopies to be paths of directed spaces gives one an $(\infty,2)$-category. So maybe that's really the next step. Which would make sense. In the fundamental $(\infty,n)$-category the $k \leq n$-homotopies should be directed, and then after that be unconstrained.

but it seems a little odd to me to allow homotopies between directed paths that pass through undirected paths along the way.

Yes, I was wondering about that, too. First my intuition also was that homotopies should be directed. But then the fact that this definition is so evident (well, ahem, I misstated it first, but that’s my fault, not the definition’s) made me rethink.

Also, given the discussion with Tim in the other thread, it seems that also requiring the homotopies to be paths of directed spaces gives one an $(\infty,2)$ -category. So maybe that’s really the next step. Which would make sense. In the fundamental $(\infty,n)$ -category the $k \leq n$ -homotopies should be directed, and then after that be unconstrained.
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeSep 30th 2010
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Author: Urs
Format: MarkdownItexI added comments to the References-section, reflecting the discussion here and in the thread on directed spaces.

I added comments to the References-section, reflecting the discussion here and in the thread on directed spaces.
- CommentRowNumber8.
- CommentAuthorEric
- CommentTimeOct 1st 2010
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Author: Eric
Format: MarkdownItexI'm really enjoying the topic du jour. Directed stuff is close to my heart. However, my conceptual guiding principle is always causality. And I always think of paths as representing the evolution of some process. When the path is directed, it means that each subsequent point is causally connected to a point before it, i.e. paths are light-like or time-like. Now, if we are to think about a path between directed paths, then I would also tend to think of this as the evolution of some process (which itself is the evolution of some other process), i.e. evolution of an evolution. I would think each point of the initial directed paths should itself trace out a directed path within the directed 2-path. In particular, if the two directed paths begin at $x$ and end at $y$, then the directed 2-path should lay entirely within the future of $x$ and the past of $y$. If you don't force your 2-paths to be "causal", then they can contain points unreachable from $x$ and that cannot reach $y$. This would seem to go against the basic motivation behind directed stuff.

I’m really enjoying the topic du jour. Directed stuff is close to my heart.

However, my conceptual guiding principle is always causality. And I always think of paths as representing the evolution of some process. When the path is directed, it means that each subsequent point is causally connected to a point before it, i.e. paths are light-like or time-like.

Now, if we are to think about a path between directed paths, then I would also tend to think of this as the evolution of some process (which itself is the evolution of some other process), i.e. evolution of an evolution. I would think each point of the initial directed paths should itself trace out a directed path within the directed 2-path.

In particular, if the two directed paths begin at $x$ and end at $y$ , then the directed 2-path should lay entirely within the future of $x$ and the past of $y$ . If you don’t force your 2-paths to be “causal”, then they can contain points unreachable from $x$ and that cannot reach $y$ . This would seem to go against the basic motivation behind directed stuff.
- CommentRowNumber9.
- CommentAuthorEric
- CommentTimeOct 1st 2010
- (edited Oct 1st 2010)
- PermaLink
Author: Eric
Format: MarkdownItexQuestion about [[fundamental (infinity,1)-category]]... >### Definition >By a **[[directed space|directed topological space]]** we here mean specifically a pair $(X,D)$ consisting of > * a [[topological space]] $X$; > * a subset $D \subset X^{[0,1]}$ of [[continuous map]]s $\gamma : [0,1] \to X$ that are are labeled as being _directed_ , > * such that > * for every orientation preserving [[homeomorphism]] $\phi : [0,1] \to [0,1]$ and every directed map $\gamma$ also $\gamma \circ \phi$ is directed; > * for any two directed paths $\gamma_1, \gamma_2$ with $\gamma_1(1) = \gamma_2(0)$ also the concatenation > $$ [0,1] \stackrel{\times 2}{\to} [0,2] \simeq [0,1] \coprod_{*} [0,1] \stackrel{(\gamma_1, \gamma_2)}{\to} X $$ > is a directed path. Would it be possible to slightly tweak this definition so that it does not explicitly depend on the continuum interval $[0,1]$? Maybe some more abstract "interval" instead? I'd be interested in (non-Hausdorff) directed topological spaces for which there is no continuum in sight. For example, you could have a finite directed topological space.
Question about fundamental (infinity,1)-category…
Definition

By a directed topological space we here mean specifically a pair $(X,D)$ consisting of
- a topological space $X$ ;
- a subset $D \subset X^{[0,1]}$ of continuous maps $\gamma : [0,1] \to X$ that are are labeled as being directed ,
- such that
  
  for every orientation preserving homeomorphism $\phi : [0,1] \to [0,1]$ and every directed map $\gamma$ also $\gamma \circ \phi$ is directed;
  
  for any two directed paths $\gamma_1, \gamma_2$ with $\gamma_1(1) = \gamma_2(0)$ also the concatenation
  $[0,1] \stackrel{\times 2}{\to} [0,2] \simeq [0,1] \coprod_{*} [0,1] \stackrel{(\gamma_1, \gamma_2)}{\to} X$
  is a directed path.
Would it be possible to slightly tweak this definition so that it does not explicitly depend on the continuum interval $[0,1]$ ? Maybe some more abstract “interval” instead? I’d be interested in (non-Hausdorff) directed topological spaces for which there is no continuum in sight. For example, you could have a finite directed topological space.
- CommentRowNumber10.
- CommentAuthorDavidRoberts
- CommentTimeOct 1st 2010
- (edited Oct 1st 2010)
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Author: DavidRoberts
Format: MarkdownItexLet $J$ be an _ordered interval_, namely a quadruple $(J,o,i,\lt)$ where $o:* \to J$ and $i:* \to J$ are maps with the pullback of $o$ and $i$ the initial object (empty set for $Top$), i.e. $o$ and $i$ are distinct, there is an endpoint preserving concatenation map $J \cup_{i,o}J \to J$, and $\lt$ is a total order with $o\lt i$. Then the definition you quote can be written down verbatim.

Let $J$ be an ordered interval, namely a quadruple $(J,o,i,\lt)$ where $o:* \to J$ and $i:* \to J$ are maps with the pullback of $o$ and $i$ the initial object (empty set for $Top$ ), i.e. $o$ and $i$ are distinct, there is an endpoint preserving concatenation map $J \cup_{i,o}J \to J$ , and $\lt$ is a total order with $o\lt i$ . Then the definition you quote can be written down verbatim.
- CommentRowNumber11.
- CommentAuthorEric
- CommentTimeOct 1st 2010
- (edited Oct 1st 2010)
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Author: Eric
Format: MarkdownItexJust reformatting in order to be able to read it better: >Let $J$ be an _ordered interval_, namely a quadruple $(J,o,i,<)$ where $o:* \to J$ and $i:* \to J$ are maps with the pullback of $o$ and $i$ the initial object (empty set for $Top$), i.e. $o$ and $i$ are distinct, there is an endpoint preserving concatenation map $J \cup_{i,o}J \to J$, and < is a total order with $o<i$. Then the definition you quote can be written down verbatim.

Just reformatting in order to be able to read it better:

Let $J$ be an ordered interval, namely a quadruple $(J,o,i,<)$ where $o:* \to J$ and $i:* \to J$ are maps with the pullback of $o$ and $i$ the initial object (empty set for $Top$ ), i.e. $o$ and $i$ are distinct, there is an endpoint preserving concatenation map $J \cup_{i,o}J \to J$ , and < is a total order with $o<i$ . Then the definition you quote can be written down verbatim.
- CommentRowNumber12.
- CommentAuthorDavidRoberts
- CommentTimeOct 1st 2010
- PermaLink
Author: DavidRoberts
Format: TextEric, you've changed the direction of my total order >:(. You'll need to change the above blue box to say i > o to keep my original meaning ;)
Eric, you've changed the direction of my total order >:(. You'll need to change the above blue box to say i > o to keep my original meaning ;)
- CommentRowNumber13.
- CommentAuthorEric
- CommentTimeOct 1st 2010
- (edited Oct 1st 2010)
- PermaLink
Author: Eric
Format: MarkdownItexOops! Will fix it :) Edit: Fixed.

Oops! Will fix it :)

Edit: Fixed.
- CommentRowNumber14.
- CommentAuthorUrs
- CommentTimeOct 1st 2010
- PermaLink
Author: Urs
Format: MarkdownItex> Would it be possible to slightly tweak this definition so that it does not explicitly depend on the continuum interval [0,1]? Just to clarify: the construction is explicitly meant to get an $(\infty,1)$-category of such paths from a directed topological space. You can consider extracting $(\infty,1)$-categories from various other gadgets by various other means, but for the definition at hand, using $[0,1]$ is a design specification.

Would it be possible to slightly tweak this definition so that it does not explicitly depend on the continuum interval [0,1]?

Just to clarify: the construction is explicitly meant to get an $(\infty,1)$ -category of such paths from a directed topological space. You can consider extracting $(\infty,1)$ -categories from various other gadgets by various other means, but for the definition at hand, using $[0,1]$ is a design specification.
- CommentRowNumber15.
- CommentAuthorTim_Porter
- CommentTimeOct 1st 2010
- PermaLink
Author: Tim_Porter
Format: MarkdownItex@Eric (No 8). ' evolution of an evolution' YES! My point is that the initial models should not assume reversibility of n-cells. That is an additional assumption that comes in when one tries to calculate (as for example groups are easier to work with than monoids). In Lyon , Philippe and Yves were using this idea to very good effect in rewriting theory. For a taste of their stuff look [here](http://arxiv.org/abs/0910.4538).

@Eric (No 8). ’ evolution of an evolution’ YES!

My point is that the initial models should not assume reversibility of n-cells. That is an additional assumption that comes in when one tries to calculate (as for example groups are easier to work with than monoids). In Lyon , Philippe and Yves were using this idea to very good effect in rewriting theory. For a taste of their stuff look here.
- CommentRowNumber16.
- CommentAuthorEric
- CommentTimeOct 1st 2010
- (edited Oct 1st 2010)
- PermaLink
Author: Eric
Format: MarkdownItexIt seems there should also be some kind of "transgression" here as well. A directed 2-path in $X$ should be a directed 1-path in "directed path space" $\vec{\mathcal{P}}X$.

It seems there should also be some kind of “transgression” here as well.

A directed 2-path in $X$ should be a directed 1-path in “directed path space” $\vec{\mathcal{P}}X$ .
- CommentRowNumber17.
- CommentAuthorUrs
- CommentTimeOct 3rd 2010
- (edited Oct 3rd 2010)
- PermaLink
Author: Urs
Format: MarkdownItexOne should now look for a proof of the following evident lemma: For $C$ any quasicategory, and $|C|$ its directed geometric realization (ordinary geometric realization with directed paths declared to be those that run from a vertex to a vertex through the 1-skeleton, preserving the natural order), there is a canonical morphism $$ C \to \Pi(|C|) $$ into the fundamental $(\infty,1)$-catgeory of $|C|$, as defined in the entry. **Conjecture** This is a weak equivalence. Meaning that: this is essentially surjective on objects and induces a weak equivlence of hom-$\infty$-groupoids. It looks like this should not be too hard to prove. But I am not sure yet. First of all, by construction we have that the morphism is even an isomorphism on objects. Then next, to compute the hom-$\infty$-groupoids, we might use the model where we hom $\Delta^k$ into $C$ with all vertices $\lt k$ restricted to land on the same object. Then it is clear that the 1-morphisms in $\Pi(|C|)$ between a given pair of objects form precisely the set of 1-morphisms in $C$, times the set of orientation-preserverving homeomorphisms $[0,1] \to [0,1]$. But we also know that there is a 2-cell in $\Pi(|C|)$ connecting any two of these reparameterized paths. So it should mean that $$ Hom_C(x,y) \to Hom_{\Pi(|C|)}(x,y) $$ is essentially surjective, i.e. an iso on $\pi_0$. So it remains to check that this is always also an iso on $\pi_k$ for $k \geq 1$. I am not yet sure how to see this precisely, but this ought to just follow from the _ordinary_ homotopy hypothesis, namely from the statement that after we invert the 1-morphisms and pass to the Kan complex $\tilde C$ we do have a weak equivalence $\tilde C \to \Pi(|\tilde C|)$. Hm....

One should now look for a proof of the following evident lemma:

For $C$ any quasicategory, and $|C|$ its directed geometric realization (ordinary geometric realization with directed paths declared to be those that run from a vertex to a vertex through the 1-skeleton, preserving the natural order), there is a canonical morphism
$C \to \Pi(|C|)$
into the fundamental $(\infty,1)$ -catgeory of $|C|$ , as defined in the entry.

Conjecture This is a weak equivalence. Meaning that: this is essentially surjective on objects and induces a weak equivlence of hom- $\infty$ -groupoids.

It looks like this should not be too hard to prove. But I am not sure yet.

First of all, by construction we have that the morphism is even an isomorphism on objects.

Then next, to compute the hom- $\infty$ -groupoids, we might use the model where we hom $\Delta^k$ into $C$ with all vertices $\lt k$ restricted to land on the same object. Then it is clear that the 1-morphisms in $\Pi(|C|)$ between a given pair of objects form precisely the set of 1-morphisms in $C$ , times the set of orientation-preserverving homeomorphisms $[0,1] \to [0,1]$ . But we also know that there is a 2-cell in $\Pi(|C|)$ connecting any two of these reparameterized paths. So it should mean that
$Hom_C(x,y) \to Hom_{\Pi(|C|)}(x,y)$
is essentially surjective, i.e. an iso on $\pi_0$ . So it remains to check that this is always also an iso on $\pi_k$ for $k \geq 1$ . I am not yet sure how to see this precisely, but this ought to just follow from the ordinary homotopy hypothesis, namely from the statement that after we invert the 1-morphisms and pass to the Kan complex $\tilde C$ we do have a weak equivalence $\tilde C \to \Pi(|\tilde C|)$ .

Hm….

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Definition