Revisions to Three questions on $\operatorname{hocolim}$

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edited Apr 13, 2017 at 12:19

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I posted this on math.stack.exchange math.stack.exchange but didn't get a helpful response, so please let me try it here.

Let $D$ be a small category and $F:D\to sSets$ a functor.

There is a bisimplicial set indicated by $$ ...\begin{array}{c}\to \\ \to\\\to\end{array}\coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0) $$ which i like to call $sres(F)_{\bullet\bullet}$. By considering either the horizontal or the vertical index first, I get two functors $H:\Delta^{op}\to sSets$ and $V:\Delta^{op}\to sSets$ (the ''vertical'' direction is the one of the simplicial sets $F(d)$, i.e. each object in the diagram above should be imagined as a vertical column).

The homotopy colimit $\operatorname{hocolim}F$ of $F$ can be defined in different ways (up to weak equivalence) and one way is $(\operatorname{hocolim}F)_n=sres(F)_{n n}$, the diagonal of $sres(F)_{\bullet\bullet}$.

Is there a weak equivalence between between $\operatorname{hocolim}F$ and $\operatorname{hocolim}H$? (I don't think there is a weak equivalence between $\operatorname{hocolim}F$ and $\operatorname{hocolim}V$, or is it?)

Let me call a simplicial set $A$ $k$-dimensional, if it is equal to its $k$-skeleton $\mathbf{sk}_k$ (If I am not mistaken, $k$ should be the smallest integer such that all elements of $A_{k+1}$ are degenerated simplices).

Suppose that the nerve $N(D)$ of $D$ is weakly equivalent to a simplicial set $A$ of dimension $k$. Is it true that $\operatorname{hocolim}F$ is weak equivalent to the homotopy colimit of the diagram $$ \coprod_{d_{k+1}\to ...\to d_0}F(d_{k+1})\begin{array}{c}\to \\ \vdots\\\to\end{array}...\begin{array}{c}\to \\ \to\\\to\end{array}\coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0), $$ (the truncated $H$) or in other words, can I ''stop'' at the $k+1$th stage of the diagram in the horizontal direction to calculate the homotopy colimit? If yes, why?

In particular, if $D$ is linear (for example $\mathbb{N}$ or $\cdot\to\cdot\to\cdot$), this would mean, that $\operatorname{hocolim}F$ is the homotopy colimit of the simple diagram $$ \coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0). $$

My third question is a little vague and like the second one but not ''for the source'' of $F$ but ''for the target''. Please don't hesitate to post an answer only to the first two questions, if this third one is not well formulated. I wondered, why one takes only two-limits of stacks and not $k$-limits. The nerve of a category is a 2-coskeletal simplicial set and this is where the reason comes from, I guess.

Suppose the functor $F$ factorize through the nerve $F:D\to CAT\xrightarrow{N} sSets$. Is it true that $\operatorname{hocolim}F$ is weakly equivalent to the homotopy colimit of the diagram $$ \coprod_{d_2\to d_1\to d_0}F(d_{2})\begin{array}{c}\to \\ \to\\\to\end{array}\coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0) $$ and what is the reason? If not, what did I misunderstand here?

I posted this on math.stack.exchange but didn't get a helpful response, so please let me try it here.

Let $D$ be a small category and $F:D\to sSets$ a functor.

There is a bisimplicial set indicated by $$ ...\begin{array}{c}\to \\ \to\\\to\end{array}\coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0) $$ which i like to call $sres(F)_{\bullet\bullet}$. By considering either the horizontal or the vertical index first, I get two functors $H:\Delta^{op}\to sSets$ and $V:\Delta^{op}\to sSets$ (the ''vertical'' direction is the one of the simplicial sets $F(d)$, i.e. each object in the diagram above should be imagined as a vertical column).

The homotopy colimit $\operatorname{hocolim}F$ of $F$ can be defined in different ways (up to weak equivalence) and one way is $(\operatorname{hocolim}F)_n=sres(F)_{n n}$, the diagonal of $sres(F)_{\bullet\bullet}$.

Is there a weak equivalence between between $\operatorname{hocolim}F$ and $\operatorname{hocolim}H$? (I don't think there is a weak equivalence between $\operatorname{hocolim}F$ and $\operatorname{hocolim}V$, or is it?)

Let me call a simplicial set $A$ $k$-dimensional, if it is equal to its $k$-skeleton $\mathbf{sk}_k$ (If I am not mistaken, $k$ should be the smallest integer such that all elements of $A_{k+1}$ are degenerated simplices).

Suppose that the nerve $N(D)$ of $D$ is weakly equivalent to a simplicial set $A$ of dimension $k$. Is it true that $\operatorname{hocolim}F$ is weak equivalent to the homotopy colimit of the diagram $$ \coprod_{d_{k+1}\to ...\to d_0}F(d_{k+1})\begin{array}{c}\to \\ \vdots\\\to\end{array}...\begin{array}{c}\to \\ \to\\\to\end{array}\coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0), $$ (the truncated $H$) or in other words, can I ''stop'' at the $k+1$th stage of the diagram in the horizontal direction to calculate the homotopy colimit? If yes, why?

In particular, if $D$ is linear (for example $\mathbb{N}$ or $\cdot\to\cdot\to\cdot$), this would mean, that $\operatorname{hocolim}F$ is the homotopy colimit of the simple diagram $$ \coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0). $$

My third question is a little vague and like the second one but not ''for the source'' of $F$ but ''for the target''. Please don't hesitate to post an answer only to the first two questions, if this third one is not well formulated. I wondered, why one takes only two-limits of stacks and not $k$-limits. The nerve of a category is a 2-coskeletal simplicial set and this is where the reason comes from, I guess.

Suppose the functor $F$ factorize through the nerve $F:D\to CAT\xrightarrow{N} sSets$. Is it true that $\operatorname{hocolim}F$ is weakly equivalent to the homotopy colimit of the diagram $$ \coprod_{d_2\to d_1\to d_0}F(d_{2})\begin{array}{c}\to \\ \to\\\to\end{array}\coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0) $$ and what is the reason? If not, what did I misunderstand here?

I posted this on math.stack.exchange but didn't get a helpful response, so please let me try it here.

Let $D$ be a small category and $F:D\to sSets$ a functor.

There is a bisimplicial set indicated by $$ ...\begin{array}{c}\to \\ \to\\\to\end{array}\coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0) $$ which i like to call $sres(F)_{\bullet\bullet}$. By considering either the horizontal or the vertical index first, I get two functors $H:\Delta^{op}\to sSets$ and $V:\Delta^{op}\to sSets$ (the ''vertical'' direction is the one of the simplicial sets $F(d)$, i.e. each object in the diagram above should be imagined as a vertical column).

The homotopy colimit $\operatorname{hocolim}F$ of $F$ can be defined in different ways (up to weak equivalence) and one way is $(\operatorname{hocolim}F)_n=sres(F)_{n n}$, the diagonal of $sres(F)_{\bullet\bullet}$.

Is there a weak equivalence between between $\operatorname{hocolim}F$ and $\operatorname{hocolim}H$? (I don't think there is a weak equivalence between $\operatorname{hocolim}F$ and $\operatorname{hocolim}V$, or is it?)

Let me call a simplicial set $A$ $k$-dimensional, if it is equal to its $k$-skeleton $\mathbf{sk}_k$ (If I am not mistaken, $k$ should be the smallest integer such that all elements of $A_{k+1}$ are degenerated simplices).

Suppose that the nerve $N(D)$ of $D$ is weakly equivalent to a simplicial set $A$ of dimension $k$. Is it true that $\operatorname{hocolim}F$ is weak equivalent to the homotopy colimit of the diagram $$ \coprod_{d_{k+1}\to ...\to d_0}F(d_{k+1})\begin{array}{c}\to \\ \vdots\\\to\end{array}...\begin{array}{c}\to \\ \to\\\to\end{array}\coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0), $$ (the truncated $H$) or in other words, can I ''stop'' at the $k+1$th stage of the diagram in the horizontal direction to calculate the homotopy colimit? If yes, why?

In particular, if $D$ is linear (for example $\mathbb{N}$ or $\cdot\to\cdot\to\cdot$), this would mean, that $\operatorname{hocolim}F$ is the homotopy colimit of the simple diagram $$ \coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0). $$

My third question is a little vague and like the second one but not ''for the source'' of $F$ but ''for the target''. Please don't hesitate to post an answer only to the first two questions, if this third one is not well formulated. I wondered, why one takes only two-limits of stacks and not $k$-limits. The nerve of a category is a 2-coskeletal simplicial set and this is where the reason comes from, I guess.

Suppose the functor $F$ factorize through the nerve $F:D\to CAT\xrightarrow{N} sSets$. Is it true that $\operatorname{hocolim}F$ is weakly equivalent to the homotopy colimit of the diagram $$ \coprod_{d_2\to d_1\to d_0}F(d_{2})\begin{array}{c}\to \\ \to\\\to\end{array}\coprod_{d_1\to d_0}F(d_1)\begin{array}{c}\to\\\to\end{array}\coprod_{d_0}F(d_0) $$ and what is the reason? If not, what did I misunderstand here?