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1. • CommentRowNumber1.
   • CommentAuthorKarol Szumiło
   • CommentTimeOct 28th 2015
   • (edited Oct 28th 2015)
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   Author: Karol Szumiło
   Format: MarkdownItexRight properness and cofibrant generation are two very desirable properties of model categories. There are plenty of interesting examples of both right proper and non-proper model categories, while most model categories of interest are cofibrantly generated. On the other hand, for many cofibrantly generated model structures, "explicit" sets of generating acyclic cofibrations are not known. Of course, the word "explicit" has no technical meaning here, I mean sets that can be somehow enumerated (not necessarily in the computational sense) rather than described by phrases like "all acyclic cofibrations between objects of cardinality bounded by...". Now, it seems to me that there is a strange coincidence between our inability to write down explicit sets of generating acyclic cofibrations and failure of right properness. Hence the following somewhat vague questions. 1. Does any one know a cofibrantly generated right proper model category with no known explicit set of generating acyclic cofibrations? 2. Does any one know a model category that is not right proper, but has an explicit set of generating acyclic cofibrations? Actually, I can give a boring answer to 1: certain injective model structures on diagrams in right proper model categories. I'm calling this a boring example since the corresponding projective model structure will have an explicit set of generating acyclic cofibrations and properness depends only on weak equivalences. Perhaps a more interesting example could be constructed using Bousfield localization. Here, I mean not the Smith style Bousfield localization, but one described e.g. in Section X.4 of Goerss--Jardine. This construction yields right proper model categories, but says nothing about cofibrant generation (although the examples I can think of at the moment happen to be explicitly cofibrantly generated). I don't know any answers to 2.

   Right properness and cofibrant generation are two very desirable properties of model categories. There are plenty of interesting examples of both right proper and non-proper model categories, while most model categories of interest are cofibrantly generated. On the other hand, for many cofibrantly generated model structures, “explicit” sets of generating acyclic cofibrations are not known. Of course, the word “explicit” has no technical meaning here, I mean sets that can be somehow enumerated (not necessarily in the computational sense) rather than described by phrases like “all acyclic cofibrations between objects of cardinality bounded by…”.

   Now, it seems to me that there is a strange coincidence between our inability to write down explicit sets of generating acyclic cofibrations and failure of right properness. Hence the following somewhat vague questions.

   1. Does any one know a cofibrantly generated right proper model category with no known explicit set of generating acyclic cofibrations?
   2. Does any one know a model category that is not right proper, but has an explicit set of generating acyclic cofibrations?

   Actually, I can give a boring answer to 1: certain injective model structures on diagrams in right proper model categories. I’m calling this a boring example since the corresponding projective model structure will have an explicit set of generating acyclic cofibrations and properness depends only on weak equivalences. Perhaps a more interesting example could be constructed using Bousfield localization. Here, I mean not the Smith style Bousfield localization, but one described e.g. in Section X.4 of Goerss–Jardine. This construction yields right proper model categories, but says nothing about cofibrant generation (although the examples I can think of at the moment happen to be explicitly cofibrantly generated).

   I don’t know any answers to 2.

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeOct 28th 2015
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   Author: Zhen Lin
   Format: MarkdownItexHere's another answer to 1. Let $\mathcal{C}$ be a locally presentable category, let $\mathcal{K}$ be a small dense full subcategory of $\mathcal{C}$, let $R : \mathcal{C} \to [\mathcal{K}^{op}, \mathbf{Set}]$ be the Yoneda representation, and let $L : [\mathcal{K}^{op}, \mathbf{Set}] \to \mathcal{C}$ be the reflector. We have the following model structure on $[\mathcal{K}^{op}, \mathbf{Set}]$: * The cofibrations are all morphisms. * The weak equivalences are the morphisms inverted by $L$. * The fibrations are the morphisms that are right orthogonal (in the strong sense!) with respect to the trivial cofibrations. This is, of course, a left Bousfield localisation of the "discrete" model structure on $[\mathcal{K}^{op}, \mathbf{Set}]$. It is right proper if and only if $L$ preserves pullbacks along fibrations. In particular, if $\mathcal{C}$ is a Grothendieck topos, then (by Giraud's theorem) we have a right proper model structure. I am not aware of any nice generating set of trivial cofibrations, but Smith's theorem applies, so for some regular cardinal $\kappa$, the set of all trivial cofibrations (= weak equivalences) between presheaves with $\lt \kappa$ elements is a generating set.

   Here’s another answer to 1.

   Let $\mathcal{C}$ be a locally presentable category, let $\mathcal{K}$ be a small dense full subcategory of $\mathcal{C}$, let $R : \mathcal{C} \to [\mathcal{K}^{op}, \mathbf{Set}]$ be the Yoneda representation, and let $L : [\mathcal{K}^{op}, \mathbf{Set}] \to \mathcal{C}$ be the reflector. We have the following model structure on $[\mathcal{K}^{op}, \mathbf{Set}]$:

   • The cofibrations are all morphisms.
   • The weak equivalences are the morphisms inverted by $L$.
   • The fibrations are the morphisms that are right orthogonal (in the strong sense!) with respect to the trivial cofibrations.

   This is, of course, a left Bousfield localisation of the “discrete” model structure on $[\mathcal{K}^{op}, \mathbf{Set}]$. It is right proper if and only if $L$ preserves pullbacks along fibrations. In particular, if $\mathcal{C}$ is a Grothendieck topos, then (by Giraud’s theorem) we have a right proper model structure. I am not aware of any nice generating set of trivial cofibrations, but Smith’s theorem applies, so for some regular cardinal $\kappa$, the set of all trivial cofibrations (= weak equivalences) between presheaves with $\lt \kappa$ elements is a generating set.

3. • CommentRowNumber3.
   • CommentAuthorChris SchommerPries
   • CommentTimeOct 29th 2015
   • (edited Oct 29th 2015)
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   Author: Chris SchommerPries
   Format: MarkdownItexLeft Bousfield localization tends to destroy right-properness, so there should be many examples of (2). Here is one important example. The complete Segal space model structure on simplicial spaces has completely explicit generating cofibrations and generating acyclic cofibrations, but it is not right proper: http://mathoverflow.net/a/40967/184

   Left Bousfield localization tends to destroy right-properness, so there should be many examples of (2).

   Here is one important example. The complete Segal space model structure on simplicial spaces has completely explicit generating cofibrations and generating acyclic cofibrations, but it is not right proper:

   http://mathoverflow.net/a/40967/184

4. • CommentRowNumber4.
   • CommentAuthorKarol Szumiło
   • CommentTimeOct 29th 2015
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   Author: Karol Szumiło
   Format: MarkdownItexThanks for an example Zhen Lin. That's something a bit unusual to me, so I will have to think about it for a while. Chris, my understanding is that left Bousfield localizations rarely come with explicit generating acyclic cofibrations even if they are localizations at explicit sets of maps. Can you actually point me to a proof showing that the obvious set of acyclic cofibrations in the Rezk model structure generates all acyclic cofibrations? I used to think that this is true, but then I realized that this set detects only so called "injective maps" that coincide with fibrations only between fibrant objects, but in order to detect general fibrations you will have to throw in a lot more stuff. At least this is what happens in the Goerss--Jardine account which is what Rezk quotes when he constructs his model structure.

   Thanks for an example Zhen Lin. That’s something a bit unusual to me, so I will have to think about it for a while.

   Chris, my understanding is that left Bousfield localizations rarely come with explicit generating acyclic cofibrations even if they are localizations at explicit sets of maps. Can you actually point me to a proof showing that the obvious set of acyclic cofibrations in the Rezk model structure generates all acyclic cofibrations? I used to think that this is true, but then I realized that this set detects only so called “injective maps” that coincide with fibrations only between fibrant objects, but in order to detect general fibrations you will have to throw in a lot more stuff. At least this is what happens in the Goerss–Jardine account which is what Rezk quotes when he constructs his model structure.

5. • CommentRowNumber5.
   • CommentAuthorCharles Rezk
   • CommentTimeOct 29th 2015
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   Author: Charles Rezk
   Format: MarkdownItexI'm not aware of explicit generating acyclic cofibrations for complete Segal spaces; the situation is as Karol says, I believe. Do *any* of the models for (infty,1)-categories have explicit generating acyclic cofibrations? I can't think of any.

   I’m not aware of explicit generating acyclic cofibrations for complete Segal spaces; the situation is as Karol says, I believe.

   Do any of the models for (infty,1)-categories have explicit generating acyclic cofibrations? I can’t think of any.

6. • CommentRowNumber6.
   • CommentAuthorKarol Szumiło
   • CommentTimeOct 29th 2015
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   Author: Karol Szumiło
   Format: MarkdownItexCharles, thank you for a confirmation. I thought that this was the case but I wasn't quite sure. I also can't think of any models of $(\infty, 1)$-categories with explicit generating acyclic cofibrations. This actually reminded me of another answer to question 1 (perhaps the most interesting one so far), the Bergner model structure on simplicial categories is right proper.

   Charles, thank you for a confirmation. I thought that this was the case but I wasn’t quite sure.

   I also can’t think of any models of $(\infty, 1)$-categories with explicit generating acyclic cofibrations. This actually reminded me of another answer to question 1 (perhaps the most interesting one so far), the Bergner model structure on simplicial categories is right proper.

7. • CommentRowNumber7.
   • CommentAuthorCharles Rezk
   • CommentTimeOct 29th 2015
   • (edited Oct 29th 2015)
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   Author: Charles Rezk
   Format: MarkdownItexHere's a proof that the "usual" acyclic cofibrations don't generate all acyclic cofibrations in CSS. Space := simplicial set. $F(n)$ := free simplicial space on one point in degree n. $E$ := discrete nerve of the walking isomorphism; $E_0 = \{x,y\}$, a discrete space with two points. There is a standard inclusion $F(1)\to E$, representing a map $x\to y$. The "usual" acyclic cofibrations $S$ are the box products of either (i) the Segal maps $G(n)\to F(n)$ or (i) the inclusion $\{x\}\to E$, with the standard simplicial cells $\partial \Delta^n\subset \Delta^n$. Here's something interesting about $S$. Given a simplical space $X$, think about $\pi_0 X_0$. Of the elements of $S$, the $only$ ones which can change $\pi_0 X_0$ when you glue them to $X$ are $\{x\}\to E$ (which adds a point to $\pi_0X_0$) and $\{x\}\times\Delta^1\cup E\times\partial\Delta^1 \to E\times \Delta^1$ (which can identify two points of $\pi_0X_0$). (The Segal maps are already bijections on $0$-spaces, and the box product of a simplicial $n$-cell with $\{x\}\to E$ attaches an $n$-cell to the $0$-space.) Let $U$ be the pushout of the diagram $$ E\amalg E \leftarrow F(1)\amalg F(1) \xrightarrow{(\langle02\rangle, \langle 13\rangle)} F(3). $$ So $U$ is the "walking 2-out-of-6 diagram". It is weakly equivalent to the terminal object in the CSS model structure, and the inclusion $\{0\}\to U$ is a CSS acyclic cofibration. I'm going to show that $\{0\}\to U$ is not in the saturation of $S$. By the small object argument, there exists a factorization $\{0\} \xrightarrow{f} A\xrightarrow{g} U$ where $f$ is in the saturation of $S$, and $g$ has the RLP with respect to $S$. Note that since $S$ consists of acyclic cofibrations, both $f$ and $g$ will be w.e. in the CSS model structure. Furthermore, if the saturation of $S$ includes *all* acyclic cofibrations, this includes $\{0\}\to U$. In this case $g$ admits a section by lifting properties. In particular, $g$ must induce a surjective map $\pi_0 A_0\to \pi_0 U=U_0=\{0,1,2,3\}$. Now we have a problem. The only way to add to $\pi_0(-)_0$ is to attach $\{x\}\to E$. But *all* maps $E\to U$ must send $E_0=\{x,y\}$ into either $\{0,2\}\subset U_0$ or $\{1,3\}\subset U_0$ (because there are points $e,e'\in E_1$ such that $(d_0e,d_1e)=(x,y)$ and $(d_0e',d_1e')=(y,x)$; however, there are no $u\in U_1$ such that $(d_0u,d_1u) \in \{(1,0), (2,1), (3,2), (3,0)\}$). This means that no factorization $\{0\}\xrightarrow{f} A\xrightarrow{g} U$ with $f$ in the saturation of $S$ can give a surjection $\pi_0A_0\to \pi_0 U$. This is a contradiction. Of course, there is a simple fix whenever you have a counterexample like this: replace $S$ with $S':=S\cup \{ (\{0\}\subset U)\square (\partial\Delta^n\subset\Delta^n),\, n\geq0 \}$. Now you get to ask the same question about $S'$ ...

   Here’s a proof that the “usual” acyclic cofibrations don’t generate all acyclic cofibrations in CSS.

   Space := simplicial set. $F(n)$ := free simplicial space on one point in degree n. $E$ := discrete nerve of the walking isomorphism; $E_0 = \{x,y\}$, a discrete space with two points. There is a standard inclusion $F(1)\to E$, representing a map $x\to y$.

   The “usual” acyclic cofibrations $S$ are the box products of either (i) the Segal maps $G(n)\to F(n)$ or (i) the inclusion $\{x\}\to E$, with the standard simplicial cells $\partial \Delta^n\subset \Delta^n$.

   Here’s something interesting about $S$. Given a simplical space $X$, think about $\pi_0 X_0$. Of the elements of $S$, the $only$ ones which can change $\pi_0 X_0$ when you glue them to $X$ are $\{x\}\to E$ (which adds a point to $\pi_0X_0$) and $\{x\}\times\Delta^1\cup E\times\partial\Delta^1 \to E\times \Delta^1$ (which can identify two points of $\pi_0X_0$). (The Segal maps are already bijections on $0$-spaces, and the box product of a simplicial $n$-cell with $\{x\}\to E$ attaches an $n$-cell to the $0$-space.)

   Let $U$ be the pushout of the diagram

   $$E\amalg E \leftarrow F(1)\amalg F(1) \xrightarrow{(\langle02\rangle, \langle 13\rangle)} F(3).$$

   So $U$ is the “walking 2-out-of-6 diagram”. It is weakly equivalent to the terminal object in the CSS model structure, and the inclusion $\{0\}\to U$ is a CSS acyclic cofibration. I’m going to show that $\{0\}\to U$ is not in the saturation of $S$.

   By the small object argument, there exists a factorization $\{0\} \xrightarrow{f} A\xrightarrow{g} U$ where $f$ is in the saturation of $S$, and $g$ has the RLP with respect to $S$. Note that since $S$ consists of acyclic cofibrations, both $f$ and $g$ will be w.e. in the CSS model structure. Furthermore, if the saturation of $S$ includes all acyclic cofibrations, this includes $\{0\}\to U$. In this case $g$ admits a section by lifting properties. In particular, $g$ must induce a surjective map $\pi_0 A_0\to \pi_0 U=U_0=\{0,1,2,3\}$.

   Now we have a problem. The only way to add to $\pi_0(-)_0$ is to attach $\{x\}\to E$. But all maps $E\to U$ must send $E_0=\{x,y\}$ into either $\{0,2\}\subset U_0$ or $\{1,3\}\subset U_0$ (because there are points $e,e'\in E_1$ such that $(d_0e,d_1e)=(x,y)$ and $(d_0e',d_1e')=(y,x)$; however, there are no $u\in U_1$ such that $(d_0u,d_1u) \in \{(1,0), (2,1), (3,2), (3,0)\}$). This means that no factorization $\{0\}\xrightarrow{f} A\xrightarrow{g} U$ with $f$ in the saturation of $S$ can give a surjection $\pi_0A_0\to \pi_0 U$. This is a contradiction.

   Of course, there is a simple fix whenever you have a counterexample like this: replace $S$ with $S':=S\cup \{ (\{0\}\subset U)\square (\partial\Delta^n\subset\Delta^n),\, n\geq0 \}$. Now you get to ask the same question about $S'$ …

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeOct 29th 2015
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   Author: Mike Shulman
   Format: MarkdownItexRegarding a model for (∞,1)-categories with explicit generating acyclic cofibrations, what about [[algebraic quasicategories]]? In the direct construction of their model structure by transfer, the generating acyclic cofibrations would be the images by the free functor of the generating acyclic cofibrations for the Joyal model structure, and hence not explicit. But I think since fibrations of algebraic quasicategories are created by the forgetful functor, every algebraic quasicategory is in particular a quasicategory, and there is an explicit set of generating acyclic cofibrations for fibrations *between fibrant objects* in the Joyal model structure, we should be able to get an explicit set of generating acyclic cofibrations for algebraic quasicategories by applying the free functor to the latter set. In general, it seems common (in particular, I think it's true in all left Bousfield localizations) to have an explicit set of generating acyclic cofibrations for fibrations between fibrant objects. So the "algebraicization" technique should apply to other nonalgebraic models, like CSS, to give other algebraic models with an explicit set of generating acyclic cofibrations. I also wonder about a model in which all objects are naturally fibrant, like topologically enriched categories. This doesn't answer the original question (2), though, since when all objects are fibrant, right properness is automatic.

   Regarding a model for (∞,1)-categories with explicit generating acyclic cofibrations, what about algebraic quasicategories? In the direct construction of their model structure by transfer, the generating acyclic cofibrations would be the images by the free functor of the generating acyclic cofibrations for the Joyal model structure, and hence not explicit. But I think since fibrations of algebraic quasicategories are created by the forgetful functor, every algebraic quasicategory is in particular a quasicategory, and there is an explicit set of generating acyclic cofibrations for fibrations between fibrant objects in the Joyal model structure, we should be able to get an explicit set of generating acyclic cofibrations for algebraic quasicategories by applying the free functor to the latter set.

   In general, it seems common (in particular, I think it’s true in all left Bousfield localizations) to have an explicit set of generating acyclic cofibrations for fibrations between fibrant objects. So the “algebraicization” technique should apply to other nonalgebraic models, like CSS, to give other algebraic models with an explicit set of generating acyclic cofibrations. I also wonder about a model in which all objects are naturally fibrant, like topologically enriched categories.

   This doesn’t answer the original question (2), though, since when all objects are fibrant, right properness is automatic.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeOct 29th 2015
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   Author: Mike Shulman
   Format: MarkdownItex(Of course, one might object that free algebraic quasicategories are themselves not very explicit. But the *set* of generators is explicit; and more importantly, by adjunction, it gives an explicit criterion for an arbitrary map in the category to be a fibration.)

   (Of course, one might object that free algebraic quasicategories are themselves not very explicit. But the set of generators is explicit; and more importantly, by adjunction, it gives an explicit criterion for an arbitrary map in the category to be a fibration.)

10. • CommentRowNumber10.
    • CommentAuthorCharles Rezk
    • CommentTimeNov 1st 2015
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    Author: Charles Rezk
    Format: MarkdownItexThat's a nice observation, Mike.

    That’s a nice observation, Mike.

11. • CommentRowNumber11.
    • CommentAuthorKarol Szumiło
    • CommentTimeNov 2nd 2015
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    Author: Karol Szumiło
    Format: MarkdownItexThat's an interesting thought, Mike. And I think it also gives a quite general recipe for "fixing" problems with Bousfield localizations we discussed above. The theorem of Nikolaus only requires a model structure where cofibrations are monomorphisms. If we have such a category and we know that the Bousfield localization at an explicit set of maps exists, then we have an explicit set of maps detecting local objects (at least if we had an explicit set detecting fibrant objects in the original category). The theorem turns them into generating acyclic cofibrations in the associated "algebraic" model structure.

    That’s an interesting thought, Mike. And I think it also gives a quite general recipe for “fixing” problems with Bousfield localizations we discussed above. The theorem of Nikolaus only requires a model structure where cofibrations are monomorphisms. If we have such a category and we know that the Bousfield localization at an explicit set of maps exists, then we have an explicit set of maps detecting local objects (at least if we had an explicit set detecting fibrant objects in the original category). The theorem turns them into generating acyclic cofibrations in the associated “algebraic” model structure.

12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeNov 2nd 2015
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    Author: Mike Shulman
    Format: MarkdownItexRight. Although it does tend to destroy other nice properties of nonalgebraic model categories, like being a presheaf category, being lccc, or having all objects cofibrant. Model category theory is like playing whack-a-mole.

    Right. Although it does tend to destroy other nice properties of nonalgebraic model categories, like being a presheaf category, being lccc, or having all objects cofibrant. Model category theory is like playing whack-a-mole.