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1. • CommentRowNumber1.
   • CommentAuthorMirco Richter
   • CommentTimeJan 19th 2013
   • (edited Jan 19th 2013)
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   Author: Mirco Richter
   Format: MarkdownItexLet $\mathbf{C}$ be an (oo,1)-cat and $X$ an object of it. Has someone already given a description of the over category $\mathbf{C}_{/X}$, in terms of categories enriched over Kan simplicial sets? I mean if we look at $\mathbf{C}$ as a category with Kan simplicial sets as hom's, how is the over category defined in this picture? The hom-set definition relays on the definition of elements $f: * \to hom(.,.)$, but in the higher context, there are not just elements $f: \Delta[0] \to hom_{\mathbf{C}}(.,.)$ but there are also the'higher elements' (higer morphisms of course) $f: \Delta[n] \to hom_{\mathbf{C}}(.,.)$, so on a first sight this is not straight forward.

   Let $\mathbf{C}$ be an (oo,1)-cat and $X$ an object of it. Has someone already given a description of the over category $\mathbf{C}_{/X}$, in terms of categories enriched over Kan simplicial sets?

   I mean if we look at $\mathbf{C}$ as a category with Kan simplicial sets as hom’s, how is the over category defined in this picture?

   The hom-set definition relays on the definition of elements $f: * \to hom(.,.)$, but in the higher context, there are not just elements $f: \Delta[0] \to hom_{\mathbf{C}}(.,.)$ but there are also the’higher elements’ (higer morphisms of course) $f: \Delta[n] \to hom_{\mathbf{C}}(.,.)$, so on a first sight this is not straight forward.

2. • CommentRowNumber2.
   • CommentAuthorMirco Richter
   • CommentTimeJan 19th 2013
   • (edited Jan 19th 2013)
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   Author: Mirco Richter
   Format: MarkdownItexIf I see it right, the idea in [[over-(infinity,1)-category]] doesn't apply 1:1 to the simplicial enriched setting. To see that, suppose $\mathbf{C}$ is an (oo,1)-category and $X,Y,Y'$ are objects of $\mathbf{C}$. Suppose the objects of $\mathbf{C}_{/X}$ are morphisms $f_1: Y \to X$. To speak of a morphism we need to know what $f_1 \in hom_{\mathbf{C}}(Y,X)$ means and in the simplicial category setting, it means $f_1 \in hom_{sSet}(\Delta[0],hom_{\mathbf{C}}(Y,X))$ Since the latter is a set '$\in$' makes sense in the usual way. The problem is then the $2$-morphism $$ \array{ Y_1 &&\stackrel{}{\to}&& Y_2 \\ & {}_{\mathllap{f_1}}\searrow &\swArrow_{f_2}& \swarrow_{\mathrlap{f'_1}} \\ && X } $$ What's that? We have $2$-morphisms $f_2 \in hom_{\mathbf{C}}(Y,X))$ and $f'_2 \in hom_{\mathbf{C}}(Y',X))$ defined by $f_2 \in hom_{sSet}(\Delta[1],hom_{\mathbf{C}}(Y,X))$ but that is different from the diagram above.

   If I see it right, the idea in over-(infinity,1)-category doesn’t apply 1:1 to the simplicial enriched setting.

   To see that, suppose $\mathbf{C}$ is an (oo,1)-category and $X,Y,Y'$ are objects of $\mathbf{C}$. Suppose the objects of $\mathbf{C}_{/X}$ are morphisms $f_1: Y \to X$. To speak of a morphism we need to know what $f_1 \in hom_{\mathbf{C}}(Y,X)$ means and in the simplicial category setting, it means

   $f_1 \in hom_{sSet}(\Delta[0],hom_{\mathbf{C}}(Y,X))$

   Since the latter is a set ’$\in$’ makes sense in the usual way.

   The problem is then the $2$-morphism

   $$\array{ Y_1 &&\stackrel{}{\to}&& Y_2 \\ & {}_{\mathllap{f_1}}\searrow &\swArrow_{f_2}& \swarrow_{\mathrlap{f'_1}} \\ && X }$$

   What’s that?

   We have $2$-morphisms $f_2 \in hom_{\mathbf{C}}(Y,X))$ and $f'_2 \in hom_{\mathbf{C}}(Y',X))$ defined by

   $f_2 \in hom_{sSet}(\Delta[1],hom_{\mathbf{C}}(Y,X))$

   but that is different from the diagram above.

3. • CommentRowNumber3.
   • CommentAuthorMirco Richter
   • CommentTimeJan 19th 2013
   • (edited Jan 19th 2013)
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   Author: Mirco Richter
   Format: MarkdownItexWhy not using the following: Objects of the slice (oo,1)-category are $n$-morphisms for any $n\in \mathbb{N}$, or to be more precise objects of $\mathbf{C}_{/X}$ are all pairs $(Y,f_n)$, where $Y \in Obj(\mathbf{C})$ is an object of $\mathbf{C}$ and $f_n: Y \to X$ is an $n$-morphism $f_n \in hom_{\mathbf{C}}(Y,X)$, which means $f_n \in hom_{sSet}(\Delta[n],hom_{\mathbf{C}}(Y,X))$ for some $n \in \mathbf{N}$. Now the arrows in the slice (oo,1)-category should be Kan simplicial sets. For any two objects $Y,Y'\in \mathbf{C}$ we have the categorical composition law $$\circ: hom_{\mathbf{C}}(Y,X) \otimes_{sSet} hom_{\mathbf{C}}(Y',Y) \to hom_{\mathbf{C}}(Y',X)$$ This way we get an arrow $(Y',f') \to (Y,f)$, which is the natural transformation of simplicial sets $t: hom_{\mathbf{C}}(Y',X) \to hom_{\mathbf{C}}(Y,X)$, restriced to the simplicial subset $f'_n(\Delta[n])$ of $hom_{\mathbf{C}}(Y',X)$. However we need a Kan simplicial set $(Y',f') \to_{\infty} (Y,f)$, but we can define that in dimension $m$ by $t: f_n'\times \Delta[m] \to f$ for $t$ as above. This is like the inner hom.

   Why not using the following:

   Objects of the slice (oo,1)-category are $n$-morphisms for any $n\in \mathbb{N}$, or to be more precise objects of $\mathbf{C}_{/X}$ are all pairs $(Y,f_n)$, where $Y \in Obj(\mathbf{C})$ is an object of $\mathbf{C}$ and $f_n: Y \to X$ is an $n$-morphism $f_n \in hom_{\mathbf{C}}(Y,X)$, which means

   $f_n \in hom_{sSet}(\Delta[n],hom_{\mathbf{C}}(Y,X))$ for some $n \in \mathbf{N}$.

   Now the arrows in the slice (oo,1)-category should be Kan simplicial sets.

   For any two objects $Y,Y'\in \mathbf{C}$ we have the categorical composition law

   $$\circ: hom_{\mathbf{C}}(Y,X) \otimes_{sSet} hom_{\mathbf{C}}(Y',Y) \to hom_{\mathbf{C}}(Y',X)$$

   This way we get an arrow $(Y',f') \to (Y,f)$, which is the natural transformation of simplicial sets $t: hom_{\mathbf{C}}(Y',X) \to hom_{\mathbf{C}}(Y,X)$, restriced to the simplicial subset $f'_n(\Delta[n])$ of $hom_{\mathbf{C}}(Y',X)$.

   However we need a Kan simplicial set $(Y',f') \to_{\infty} (Y,f)$, but we can define that in dimension $m$ by $t: f_n'\times \Delta[m] \to f$ for $t$ as above.

   This is like the inner hom.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 21st 2013
   • (edited Jan 21st 2013)
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   Author: Urs
   Format: MarkdownItexHard to tell for me what you are asking here. Maybe the quickest way to get a handle on $\infty$-slices as Kan-complex enriched categories is this: if you obtain your Kan-complex enriched category from a model category, then the Kan-complex slice category is analogously given by the [[model structure on an over category|slice model structure]].

   Hard to tell for me what you are asking here.

   Maybe the quickest way to get a handle on $\infty$-slices as Kan-complex enriched categories is this: if you obtain your Kan-complex enriched category from a model category, then the Kan-complex slice category is analogously given by the slice model structure.

5. • CommentRowNumber5.
   • CommentAuthorMirco Richter
   • CommentTimeJan 21st 2013
   • (edited Jan 22nd 2013)
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   Author: Mirco Richter
   Format: MarkdownItexHmm I thought to proof that this a valid slice category presentation, I should apply Cordiers simplicial nerve functor and then see, if what we get is equivalent to his standard definition of the slice in terms of quasy categories.

   Hmm I thought to proof that this a valid slice category presentation, I should apply Cordiers simplicial nerve functor and then see, if what we get is equivalent to his standard definition of the slice in terms of quasy categories.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJan 22nd 2013
   • (edited Jan 22nd 2013)
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   Author: Urs
   Format: MarkdownItexWhat is "this" in your last message? The proof that the slice model structure presensent the slice $\infty$-category (if we are slicing over a fibrant representative) is spelled out at _[[slice model structure]]_. Or are you after the description of the slice as a quasi-category? That is simply this: if $\mathcal{C}$ is a quasi-category and $X$ an object, then then $n$-cells of $\mathcal{C}_{/X}$ are those maps $\Delta^n \ast \Delta^0 \to \mathcal{C}$ out of the [[join of simplicial sets]] that take the cone point $\Delta^0$ to $X$.

   What is “this” in your last message?

   The proof that the slice model structure presensent the slice $\infty$-category (if we are slicing over a fibrant representative) is spelled out at slice model structure.

   Or are you after the description of the slice as a quasi-category? That is simply this: if $\mathcal{C}$ is a quasi-category and $X$ an object, then then $n$-cells of $\mathcal{C}_{/X}$ are those maps $\Delta^n \ast \Delta^0 \to \mathcal{C}$ out of the join of simplicial sets that take the cone point $\Delta^0$ to $X$.

7. • CommentRowNumber7.
   • CommentAuthorMirco Richter
   • CommentTimeJan 22nd 2013
   • (edited Jan 22nd 2013)
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   Author: Mirco Richter
   Format: MarkdownItex*this* means my proposed definition in post #3, because of what I said in #2 ... The last part of your last post is clear. This is part of HTT, but not what I'm after. I'm talking about: data: Category enriched over Kan simplicial sets (This is a prsentation of an (oo,1)-category) Question: Given an object of that category, what is the slice category over that object. Can't see why we have to take model structures into consideration. Most likely there is a bug in our communication, somewhere...

   this means my proposed definition in post #3, because of what I said in #2 …

   The last part of your last post is clear. This is part of HTT, but not what I’m after.

   I’m talking about:

   data:

   Category enriched over Kan simplicial sets (This is a prsentation of an (oo,1)-category)

   Question:

   Given an object of that category, what is the slice category over that object.

   Can’t see why we have to take model structures into consideration.

   Most likely there is a bug in our communication, somewhere…

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 22nd 2013
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   Author: Urs
   Format: MarkdownItexI am not aware of a direct formula for a general Kan-complex enriched category. But I said: in case it happens to come to you as the full subcategory of fibrant-cofibrant objects of a simplicial model category, then the slicing is given simply as the ordinary slicing of the model structure over a fibrant representative objects. If your Kan-complex enriched category does not arise this way, you can in principle 1. apply the homotopy coherent nerve to obtain the coresponding quasi-category, then 2. produce its slice as above and then 3. turn that again into a Kan complex enriched category by the adjoint of the homotopy coherent nerve.

   I am not aware of a direct formula for a general Kan-complex enriched category.

   But I said: in case it happens to come to you as the full subcategory of fibrant-cofibrant objects of a simplicial model category, then the slicing is given simply as the ordinary slicing of the model structure over a fibrant representative objects.

   If your Kan-complex enriched category does not arise this way, you can in principle 1. apply the homotopy coherent nerve to obtain the coresponding quasi-category, then 2. produce its slice as above and then 3. turn that again into a Kan complex enriched category by the adjoint of the homotopy coherent nerve.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJan 22nd 2013
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   Author: Mike Shulman
   Format: MarkdownItexHere's a more abstract idea: slice categories are a special case of comma categories, which are a weighted limit. Now consider the corresponding weighted *homotopy* limit in the model category of simplicially enriched categories. This might not work, however, if the latter model category is insufficiently well-behaved --- e.g. it is not a monoidal model structure, hence not model-wise enriched over itself, and so it's not entirely clear what sort of "weighting" one would need to consider. But it's perhaps a direction worth thinking about.

   Here’s a more abstract idea: slice categories are a special case of comma categories, which are a weighted limit. Now consider the corresponding weighted homotopy limit in the model category of simplicially enriched categories. This might not work, however, if the latter model category is insufficiently well-behaved — e.g. it is not a monoidal model structure, hence not model-wise enriched over itself, and so it’s not entirely clear what sort of “weighting” one would need to consider. But it’s perhaps a direction worth thinking about.