4
$\begingroup$

The construction of homotopy limit in Goerss and Jardin (Simplicial homotopy theory [pp. 400-401]) is as follows: Let $X :\mathcal{I}\rightarrow \mathcal{S}$ be a small diagram of simplicial sets. Construct a cosimplicial object $T^{\bullet}X$ in $\mathcal{S}^{\mathcal{I}}$ with $n$-cosimplices

$${\big(T^{\bullet}X\big)}^n(j)=\prod_{{j\rightarrow i_{ \tiny 0}\rightarrow\cdots \rightarrow i_{\tiny {n}}}}X(i_n).$$ Take a pointwise fibrant replacement $Y$ of $X$ then the homotopy limit is $$\underset{\substack{\longleftarrow \\ \mathcal{I}}}{holim}\ X=Tot\ \underset{\substack{\longleftarrow \\ \mathcal{I}}}{lim}\ T^{\bullet}Y.$$ Where the totalization functor is, $Tot:c\mathcal{S}\rightarrow \mathcal{S}$ as the equalizer of the following diagram $$ Tot(X)\longrightarrow \prod_{ n\geq 0}Hom(\it{\Delta}^{n}, X^n)\ \ \substack{\longrightarrow\\\longrightarrow}\prod_{\phi:[n]\rightarrow[m]} Hom(\it{\Delta}^{m}, X^m)$$

where $Hom(\_\ ,\_)$ is the internal hom in $c\mathcal{S}.$

Now I want construct homotopy limits of bisimplicial sets just by repalcing simplicial sets in the above definition by bisimplicial sets . My question is, is this definiton correct for homotopy limit of bisimplicial sets. Precisely, here is what I do:

Let $X :\mathcal{I}\rightarrow \mathcal{S}^2$ be a small diagram of bisimplicial sets. Construct a cosimplicial object $T^{\bullet}X$ in ${\mathcal{S}^2}^{\mathcal{I}}$ with $n$-cosimplices

$${\big(T^{\bullet}X\big)}^n(j)=\prod_{{j\rightarrow i_{ \tiny 0}\rightarrow\cdots \rightarrow i_{\tiny {n}}}}X(i_n).$$ Take a pointwise fibrant replacement $Y$ of $X$ then the homotopy limit is $$\underset{\substack{\longleftarrow \\ \mathcal{I}}}{holim}\ X=Tot\ \underset{\substack{\longleftarrow \\ \mathcal{I}}}{lim}\ T^{\bullet}Y.$$ Where the totalization functor is, $Tot:c\mathcal{S}^2\rightarrow \mathcal{S}^2$ as the equalizer of the following diagram $$ Tot(X)\longrightarrow \prod_{ n\geq 0}Hom(\it{\Delta}^{n,n}, X^n)\ \ \substack{\longrightarrow\\\longrightarrow}\prod_{\phi:[n]\rightarrow[m]} Hom(\it{\Delta}^{m,m}, X^m)$$

where $Hom(\_\ ,\_)$ is the internal hom in $c\mathcal{S}^2.$

Does this really give homotopy limit of bisimplicial sets.

Edit: The weak equivalences in $\mathcal {S^2}$ are pointwise weak equivalences, i.e. a morphism of bisimplicial sets $X\rightarrow Y$ is a weak equivalence if $X_{n} \rightarrow Y_{n}$ is a weak equivalence of simplicial sets for all $n$, where $X_n:=X(n,*)$

$\endgroup$
  • 2
    $\begingroup$ It's impossible to answer this until we know what you are using as weak equivalences of bisimplicial sets. $\endgroup$ – Charles Rezk Feb 7 '17 at 13:59
  • 1
    $\begingroup$ Sorry, the weak equivalences are pointwise weak equivalences. $\endgroup$ – Girish Feb 8 '17 at 12:45
  • $\begingroup$ I am confused by what you mean with "pointwise equivalences". Aren't $X_{n,m}$ and $Y_{n,m}$ just sets? $\endgroup$ – Denis Nardin Feb 9 '17 at 13:12
  • $\begingroup$ @Denis Nardin Please see the edit. $\endgroup$ – Girish Feb 9 '17 at 14:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.