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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 12th 2010
   • (edited Dec 12th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThis is a follow up to the thread about the [[Lawvere interval]]. I realized that things weren't as easy as they seemed, so I e-mailed Cisinski again. (The claim is that the set of __anodyne pregenerators__ relative to the cylinder defined by the [[Lawvere interval]] of the class of right anodyne morphisms consists of the single morphism $\{1\}\hookrightarrow \Delta^1$ (indeed, the the idea works for _any_ injective interval object, whence we may use the nerve of the contractible groupoid with two distinct objects). (Please be aware that "anodyne pregenerators" is not standard terminology. These are often just called "generators", but that word is extremely overloaded in this context, so I have taken it upon myself to coin a new term.) For a look at how you generate a set of generators from a set of pregenerators and a set of generating cofibrations, either look at the construction 1.3.12 of Cisinski's book, or my recent post, which has the construction practically word-for-word). Since I'm sure that some of you (specifically Urs was interested) would like an explanation, I reproduce below the relevant fragment of Cisinski's e-mail (along with my quoted question): >>I was thinking about your last e-mail earlier this week, and I realized that I'm not exactly sure why l(r(Λ_J({{1}->Δ[1]})) (where J=(J,j^0,j^1) is the separated segment defined by the nerve of the contractible groupoid with two objects) is actually equivalent to the class of right anodyne maps (in the special case you noted). >First, you should consider the description of the class of right anodyne maps given for instance by Lemma 6.6 in the paper arXiv:0902.1954 (this result may be attributed to Gabriel and Zisman though; see the references in the `proof' of loc. cit.). On the other hand, any right fibration p:X-->Y between quasi-categories has the right lifting property with respect to the smallest class of anodyne extensions with respect to the interval J (see Prop. 2.7 in Joyal's paper Quasi-categories and Kan complexes, J. Pure Appl. Alg. 175 (2002), 207-222). But this implies that this remains true without the assumption that Y is a quasi-category: this is because, if you want to check the right lifting property of p:X-->Y with respect to a map i:A-->B, for any maps a:A-->X and b:B-->Y such that pa=bi, you may replace p by its pullback along b, and thus always assume that B=Y. In our case, B will always be the nerve of a small category (whence a quasi-category). Therefore, you will get that any right fibration is a fibration for the minimal model structure (over any simplicial set). Using the good behaviour of right anodyne extensions with respect to the cartesian product (the result of Gabriel and Zisman I first mentionned), it is easy to conclude. Editorial comment: The content of the lemma in the paper arXiv:0902.1954 is the well-known lemma that the class of right anodyne morphisms is generated (not pre-generated, but "generated as a weakly saturated class of morphisms") by the set of smash products $\iota^n\wedge \{1\}$ where $\iota^n:\partial\Delta^n\hookrightarrow \Delta^n$ is the canonical inclusion, and $\{1\}:\Delta^0\to \Delta^1$ is the inclusion of the terminal vertex. The other relevant result (from Joyal's paper) is the one that shows that right fibrations between quasicategories are quasifibrations and conservative.

   This is a follow up to the thread about the Lawvere interval.

   I realized that things weren’t as easy as they seemed, so I e-mailed Cisinski again. (The claim is that the set of anodyne pregenerators relative to the cylinder defined by the Lawvere interval of the class of right anodyne morphisms consists of the single morphism $\{1\}\hookrightarrow \Delta^1$ (indeed, the the idea works for any injective interval object, whence we may use the nerve of the contractible groupoid with two distinct objects). (Please be aware that “anodyne pregenerators” is not standard terminology. These are often just called “generators”, but that word is extremely overloaded in this context, so I have taken it upon myself to coin a new term.) For a look at how you generate a set of generators from a set of pregenerators and a set of generating cofibrations, either look at the construction 1.3.12 of Cisinski’s book, or my recent post, which has the construction practically word-for-word).

   Since I’m sure that some of you (specifically Urs was interested) would like an explanation, I reproduce below the relevant fragment of Cisinski’s e-mail (along with my quoted question):

   I was thinking about your last e-mail earlier this week, and I realized that I’m not exactly sure why l(r(Λ_J({{1}->Δ[1]})) (where J=(J,j^0,j^1) is the separated segment defined by the nerve of the contractible groupoid with two objects) is actually equivalent to the class of right anodyne maps (in the special case you noted).

   First, you should consider the description of the class of right anodyne maps given for instance by Lemma 6.6 in the paper arXiv:0902.1954 (this result may be attributed to Gabriel and Zisman though; see the references in the ‘proof’ of loc. cit.). On the other hand, any right fibration p:X–>Y between quasi-categories has the right lifting property with respect to the smallest class of anodyne extensions with respect to the interval J (see Prop. 2.7 in Joyal’s paper Quasi-categories and Kan complexes, J. Pure Appl. Alg. 175 (2002), 207-222). But this implies that this remains true without the assumption that Y is a quasi-category: this is because, if you want to check the right lifting property of p:X–>Y with respect to a map i:A–>B, for any maps a:A–>X and b:B–>Y such that pa=bi, you may replace p by its pullback along b, and thus always assume that B=Y. In our case, B will always be the nerve of a small category (whence a quasi-category). Therefore, you will get that any right fibration is a fibration for the minimal model structure (over any simplicial set). Using the good behaviour of right anodyne extensions with respect to the cartesian product (the result of Gabriel and Zisman I first mentionned), it is easy to conclude.

   Editorial comment: The content of the lemma in the paper arXiv:0902.1954 is the well-known lemma that the class of right anodyne morphisms is generated (not pre-generated, but “generated as a weakly saturated class of morphisms”) by the set of smash products $\iota^n\wedge \{1\}$ where $\iota^n:\partial\Delta^n\hookrightarrow \Delta^n$ is the canonical inclusion, and $\{1\}:\Delta^0\to \Delta^1$ is the inclusion of the terminal vertex.

   The other relevant result (from Joyal’s paper) is the one that shows that right fibrations between quasicategories are quasifibrations and conservative.

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 14th 2010
   • (edited Dec 14th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexI think that the key point that Cisinski did not explain in his e-mail (although it follows trivially from a theorem in his book) is the following fact: [My question and subsequent answer on MO](http://mathoverflow.net/questions/49210/abstract-classes-of-anodyne-maps-relative-to-an-interval-in-a-presheaf-category/49241#49241) (Well, at least it's good to know that the proof of 1.3.58, which implies the result, ended up being rather sophisticated. I tried proving it using elementary approaches, but this, alas, was not to be.) So the formula is: Fix any choice of injective cylinder object $I$ and pick a set of monomorphisms $S$ such that $LLP(RLP(S\wedge Cof))$ is your desired class of anodyne extensions (where $S\wedge Cof$ is defined to be the class of morphisms of the form $f\wedge g$ where $f\in S$ and $g\in Cof$. If $M$ is a cellular model for the class of monomorphisms (i.e. a set $M$ admitting the small object argument and such that $LLP(RLP(M))=Cof$, then $An(S):=LLP(RLP(S\wedge Cof))=LLP(RLP(S\wedge M))=LLP(RLP(\Lambda_I(S,M))$. This is extremely nice to know, since first of all, it gives us a really easy way to prove the existence of a whole bunch of model structures. Further, it seems like many of the interesting examples that come up in practice can be described in terms of finitely many morphisms. For instance, the classical model structure on simplicial sets is pregenerated by the pair of inclusions $\{i\}:\Delta^0\hookrightarrow \Delta^1$ for $i\in\{0,1\}$. The model structures for right and left fibrations are the case for $i=0$ or $i=1$. The Joyal model structure can be described as being pregenerated by the single morphism $\iota:\Lambda^2_1 \hookrightarrow \Delta^2$. I think that there's a kernel of something interesting here, but I'm not yet sure what.

   I think that the key point that Cisinski did not explain in his e-mail (although it follows trivially from a theorem in his book) is the following fact:

   My question and subsequent answer on MO

   (Well, at least it’s good to know that the proof of 1.3.58, which implies the result, ended up being rather sophisticated. I tried proving it using elementary approaches, but this, alas, was not to be.)

   So the formula is: Fix any choice of injective cylinder object $I$ and pick a set of monomorphisms $S$ such that $LLP(RLP(S\wedge Cof))$ is your desired class of anodyne extensions (where $S\wedge Cof$ is defined to be the class of morphisms of the form $f\wedge g$ where $f\in S$ and $g\in Cof$. If $M$ is a cellular model for the class of monomorphisms (i.e. a set $M$ admitting the small object argument and such that $LLP(RLP(M))=Cof$, then $An(S):=LLP(RLP(S\wedge Cof))=LLP(RLP(S\wedge M))=LLP(RLP(\Lambda_I(S,M))$. This is extremely nice to know, since first of all, it gives us a really easy way to prove the existence of a whole bunch of model structures.

   Further, it seems like many of the interesting examples that come up in practice can be described in terms of finitely many morphisms.

   For instance, the classical model structure on simplicial sets is pregenerated by the pair of inclusions $\{i\}:\Delta^0\hookrightarrow \Delta^1$ for $i\in\{0,1\}$. The model structures for right and left fibrations are the case for $i=0$ or $i=1$.

   The Joyal model structure can be described as being pregenerated by the single morphism $\iota:\Lambda^2_1 \hookrightarrow \Delta^2$.

   I think that there’s a kernel of something interesting here, but I’m not yet sure what.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 14th 2010
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   Author: Urs
   Format: MarkdownItexHary, just have a split second to look at your message. Remind me, when you talk about the model structure for left and right fibs, is it over a general base simplicial set or just over the point?

   Hary, just have a split second to look at your message. Remind me, when you talk about the model structure for left and right fibs, is it over a general base simplicial set or just over the point?

4. • CommentRowNumber4.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 14th 2010
   • (edited Dec 14th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThis holds in full generality over any base. This method of giving an anodyne class of maps is really powerful, because it is equivalent to giving a compatible family of choices of fibrant objects for the model structure over any base (by a theorem of Joyal, the cofibrations and fibrant objects classify the model category structure (provided it exists) up to equality). However, a homotopical structure (this is what Cisinski calls a pair consisting of a cylinder and a class of anodynes for that cylinder) specifies more than a model structure. It specifies a compatible family of model structures over each object. This is more powerful than just specifying a model structure, since there can be many different "local" model structures that patch together to form the same global one. The fact that model structures do not localize uniquely (here, by localize, I mean with respect to slicing over an object) is definitely evidence that the current notion of a model structure is really "wrong", at least in categories with appreciable "local" structure, like toposes). Here's another enlightening excerpt from the e-mail he sent me on the "failure" of the completeness of a homotopical structure: > However, there is something very interesting here: in general, this homotopy structure is not complete in the sense of Definition 1.3.48 of my book. This means that, whenever one considers an object X, the homotopy structure defined by J and by the class of right I-anodyne extensions over X (see Proposition 1.3.53 in loc. cit.) property (b) of Proposition 1.3.61 in loc. cit. fails in general (note that this failure may happen only when X is not fibrant; this follows easily from Corollary 1.3.35 and from Scholium 1.3.45). But this is far from being a flaw. For instance, consider a quasi-category X which is not a Kan complex. Then the model structure on the category of simplicial sets associated to the homotopy structure defined by J and the class of right I-anodyne extensions over X is very interesting: its fibrant objects are the right fibrations Y-->X. But right fibrations Y-->X correspond precisely to the (infinity,1)-category version of the notion of Grothendieck fibration whose fibers are groupoids (recall that the fibers of any right fibration are Kan complexes). In other words, we obtain the homotopy theory of (pre)stacks of infinity-groupoids over X (see also section 2.1.4 in Lurie's book for the dual version (with left fibrations), as well as Proposition 2.4.2.4. in Lurie's book). In this case the failure of property (b) of Proposition 1.3.61 of my book comes from the fact that there are functors between fibred (infinity,1)-categories over X which are not cartesian. On the other hand, the fact that, in the case where X is the point, we just get the usual model structure, may thus be interpreted as the fact that the category of stacks of infinity-groupoids over the point is precisely the category of infinity-groupoids.

   This holds in full generality over any base. This method of giving an anodyne class of maps is really powerful, because it is equivalent to giving a compatible family of choices of fibrant objects for the model structure over any base (by a theorem of Joyal, the cofibrations and fibrant objects classify the model category structure (provided it exists) up to equality).

   However, a homotopical structure (this is what Cisinski calls a pair consisting of a cylinder and a class of anodynes for that cylinder) specifies more than a model structure. It specifies a compatible family of model structures over each object. This is more powerful than just specifying a model structure, since there can be many different “local” model structures that patch together to form the same global one. The fact that model structures do not localize uniquely (here, by localize, I mean with respect to slicing over an object) is definitely evidence that the current notion of a model structure is really “wrong”, at least in categories with appreciable “local” structure, like toposes).

   Here’s another enlightening excerpt from the e-mail he sent me on the “failure” of the completeness of a homotopical structure:

   However, there is something very interesting here: in general, this homotopy structure is not complete in the sense of Definition 1.3.48 of my book. This means that, whenever one considers an object X, the homotopy structure defined by J and by the class of right I-anodyne extensions over X (see Proposition 1.3.53 in loc. cit.) property (b) of Proposition 1.3.61 in loc. cit. fails in general (note that this failure may happen only when X is not fibrant; this follows easily from Corollary 1.3.35 and from Scholium 1.3.45). But this is far from being a flaw. For instance, consider a quasi-category X which is not a Kan complex. Then the model structure on the category of simplicial sets associated to the homotopy structure defined by J and the class of right I-anodyne extensions over X is very interesting: its fibrant objects are the right fibrations Y–>X. But right fibrations Y–>X correspond precisely to the (infinity,1)-category version of the notion of Grothendieck fibration whose fibers are groupoids (recall that the fibers of any right fibration are Kan complexes). In other words, we obtain the homotopy theory of (pre)stacks of infinity-groupoids over X (see also section 2.1.4 in Lurie’s book for the dual version (with left fibrations), as well as Proposition 2.4.2.4. in Lurie’s book). In this case the failure of property (b) of Proposition 1.3.61 of my book comes from the fact that there are functors between fibred (infinity,1)-categories over X which are not cartesian. On the other hand, the fact that, in the case where X is the point, we just get the usual model structure, may thus be interpreted as the fact that the category of stacks of infinity-groupoids over the point is precisely the category of infinity-groupoids.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 14th 2010
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   Author: Urs
   Format: MarkdownItexOkay, so when you say > For instance, the classical model structure on simplicial sets is pregenerated by the pair of inclusions $\{i\}:\Delta^0\hookrightarrow \Delta^1$ for $i\in\{0,1\}$. The model structures for right and left fibrations are the case for $i=0$ or $i=1$. what do you mean over a general base?

   Okay, so when you say

   For instance, the classical model structure on simplicial sets is pregenerated by the pair of inclusions $\{i\}:\Delta^0\hookrightarrow \Delta^1$ for $i\in\{0,1\}$. The model structures for right and left fibrations are the case for $i=0$ or $i=1$.

   what do you mean over a general base?

6. • CommentRowNumber6.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 14th 2010
   • (edited Dec 14th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexCommutative triangles $\Delta^0\hookrightarrow \Delta^1 \to X$. Here, let me link you to the page where it's discussed (1.3.45). Hold on. [Clicky!](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf#Page=65) That's a discussion of the general case.

   Commutative triangles $\Delta^0\hookrightarrow \Delta^1 \to X$.

   Here, let me link you to the page where it’s discussed (1.3.45). Hold on.

   Clicky!

   That’s a discussion of the general case.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeDec 14th 2010
   • (edited Dec 14th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. I need to find a free minute to look into this stuff in a little bit more detail than I can right now. Maybe over the weekend.

   Thanks. I need to find a free minute to look into this stuff in a little bit more detail than I can right now. Maybe over the weekend.