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Mapping spaces for pro-objects in a simplicial model category

How does hocolim relate to Hom?The two-sided simplicial bar construction is Reedy-cofibrantHomotopy limitsConstructing model category from given categoryA construction with homotopy colimits and homotopy pullbacks for descent.Lifting a homotopy class $S^kto X$ into a simplicial set $X$ which is not fibrant but satisfies some weaker horn filling conditionEnriching categories of simplicial objectsDo Homotopy limits commute with right Quillen functorsUnderlying quasicategory of a model category through framings?map on connected components is injective

0

$begingroup$

If $C$ is a simplicial model category (I have simplicial commutative rings in mind), then one can simplicially enrich pro-$C$ over sSet by taking $$lim_j mathrmcolim_i underlinemathrmHom_C(A_i,B_j)$$ for any two pro-systems $A_i$ and $B_j$. Note $underlinemathrmHom$ here denotes the enrichment. These are indexed over small cofiltered indexing categories. This is for example done by Isaksen when he defines a model structure on pro-$C$ given that $C$ is proper.

My question is why this isn't $$mathrmholim_jmathrmhocolim_iunderlinemathrmHom_C(A_i,B_j)
.$$ Homotopy colimits over filtered indexing categories are presented by the strict colimit (maybe you need some condition on $C$ like combinatorial), so this doesn't cause an issue, but what about the homotopy limit?

For example, Galatius and Venkatesh show in their paper "Derived Galois Deformation Rings" that the natural map lim $to$ holim in the above definition is a weak equivalence (in the Quillen model structure) of simplicial sets if

1. 
   $J$ is cofinite (i.e. $j : j < j_0$ is finite for all $j_0 in J$),


2. each $B_j$ is cofibrant, and


3. The maps $B_j_0 to lim_j<j_0 B_j$ are fibrations for all $j_0$.

In Isaksen's language this means that $B_j$ is strongly fibrant.

But since these conditions on $J$ and $B_j$ are not always satisfied (well, the first one is, without loss of generality), I wouldn't expect holim to give lim all this time. So is there some conceptual explanation of why one would take lim instead of holim in the definition?

algebraic-topology category-theory homotopy-theory simplicial-stuff model-categories

share|cite|improve this question

edited Mar 17 at 13:17

Ashwin Iyengar

asked Mar 5 at 19:38

Ashwin IyengarAshwin Iyengar

1,181615

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I mean, you're never going to get a strictly associative composition using homotopy limits for your hom-spaces. I wouldn't be surprised if you can use a definition like that in an $infty$-categorical framework.
  $endgroup$
  – Kevin Carlson
  Mar 6 at 1:39
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ahh good point, hadn't thought about that.
  $endgroup$
  – Ashwin Iyengar
  Mar 6 at 11:06
  

  
  
  
  

add a comment | 

0

$begingroup$

If $C$ is a simplicial model category (I have simplicial commutative rings in mind), then one can simplicially enrich pro-$C$ over sSet by taking $$lim_j mathrmcolim_i underlinemathrmHom_C(A_i,B_j)$$ for any two pro-systems $A_i$ and $B_j$. Note $underlinemathrmHom$ here denotes the enrichment. These are indexed over small cofiltered indexing categories. This is for example done by Isaksen when he defines a model structure on pro-$C$ given that $C$ is proper.

My question is why this isn't $$mathrmholim_jmathrmhocolim_iunderlinemathrmHom_C(A_i,B_j)
.$$ Homotopy colimits over filtered indexing categories are presented by the strict colimit (maybe you need some condition on $C$ like combinatorial), so this doesn't cause an issue, but what about the homotopy limit?

For example, Galatius and Venkatesh show in their paper "Derived Galois Deformation Rings" that the natural map lim $to$ holim in the above definition is a weak equivalence (in the Quillen model structure) of simplicial sets if

1. 
   $J$ is cofinite (i.e. $j : j < j_0$ is finite for all $j_0 in J$),


2. each $B_j$ is cofibrant, and


3. The maps $B_j_0 to lim_j<j_0 B_j$ are fibrations for all $j_0$.

In Isaksen's language this means that $B_j$ is strongly fibrant.

But since these conditions on $J$ and $B_j$ are not always satisfied (well, the first one is, without loss of generality), I wouldn't expect holim to give lim all this time. So is there some conceptual explanation of why one would take lim instead of holim in the definition?

algebraic-topology category-theory homotopy-theory simplicial-stuff model-categories

share|cite|improve this question

edited Mar 17 at 13:17

Ashwin Iyengar

asked Mar 5 at 19:38

Ashwin IyengarAshwin Iyengar

1,181615

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I mean, you're never going to get a strictly associative composition using homotopy limits for your hom-spaces. I wouldn't be surprised if you can use a definition like that in an $infty$-categorical framework.
  $endgroup$
  – Kevin Carlson
  Mar 6 at 1:39
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ahh good point, hadn't thought about that.
  $endgroup$
  – Ashwin Iyengar
  Mar 6 at 11:06
  

  
  
  
  

add a comment | 

$begingroup$

If $C$ is a simplicial model category (I have simplicial commutative rings in mind), then one can simplicially enrich pro-$C$ over sSet by taking $$lim_j mathrmcolim_i underlinemathrmHom_C(A_i,B_j)$$ for any two pro-systems $A_i$ and $B_j$. Note $underlinemathrmHom$ here denotes the enrichment. These are indexed over small cofiltered indexing categories. This is for example done by Isaksen when he defines a model structure on pro-$C$ given that $C$ is proper.

My question is why this isn't $$mathrmholim_jmathrmhocolim_iunderlinemathrmHom_C(A_i,B_j)
.$$ Homotopy colimits over filtered indexing categories are presented by the strict colimit (maybe you need some condition on $C$ like combinatorial), so this doesn't cause an issue, but what about the homotopy limit?

For example, Galatius and Venkatesh show in their paper "Derived Galois Deformation Rings" that the natural map lim $to$ holim in the above definition is a weak equivalence (in the Quillen model structure) of simplicial sets if

1. 
   $J$ is cofinite (i.e. $j : j < j_0$ is finite for all $j_0 in J$),


2. each $B_j$ is cofibrant, and


3. The maps $B_j_0 to lim_j<j_0 B_j$ are fibrations for all $j_0$.

In Isaksen's language this means that $B_j$ is strongly fibrant.

But since these conditions on $J$ and $B_j$ are not always satisfied (well, the first one is, without loss of generality), I wouldn't expect holim to give lim all this time. So is there some conceptual explanation of why one would take lim instead of holim in the definition?

algebraic-topology category-theory homotopy-theory simplicial-stuff model-categories

share|cite|improve this question

edited Mar 17 at 13:17

Ashwin Iyengar

asked Mar 5 at 19:38

Ashwin IyengarAshwin Iyengar

1,181615

$endgroup$

share|cite|improve this question

edited Mar 17 at 13:17

Ashwin Iyengar

asked Mar 5 at 19:38

Ashwin IyengarAshwin Iyengar

1,181615

share|cite|improve this question

edited Mar 17 at 13:17

Ashwin Iyengar

asked Mar 5 at 19:38

Ashwin IyengarAshwin Iyengar

1,181615

asked Mar 5 at 19:38

Ashwin IyengarAshwin Iyengar

1,181615

asked Mar 5 at 19:38

Ashwin IyengarAshwin Iyengar

1,181615