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If matching maps of a cosimplicial space are fibrations up to degree $n$, does the $n$-th partial totalization have the 'correct' homotopy type?

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2

$begingroup$

Let $X$ be a cosimplicial space and suppose that the matching maps:
$$
s:X^krightarrow M^k-1X
$$
are fibrations for all $kleq n$. On one hand we can form the partial totalization $Tot_n(X)$, on the other hand we can take the fibrant replacement $RX$ and form $widetildeTot_n(X):=Tot_n(RX)$.

Do we have $Tot_n(X)simeq widetildeTot_n(X)$?

In other words, is the partial totalization in this case already homotopy invariant?

For example, totalization preserves weak equivalences between Reedy fibrant cosimplicial spaces (i.e. those for which all the matching maps are fibrations): if $X$ and $Y$ are (levelwise) weakly equivalent and both are Ready fibrant, then $Tot(X)simeq Tot(Y)$.

I guess I am asking if there is an analogous statement for the case of partial totalizations, and would it be as simple as I suggested - partial totalization preserves weak equivalences between truncated cosimplicial spaces if they are ''fibrant up to appropriate degree''.

algebraic-topology homotopy-theory simplicial-stuff monoidal-categories

share|cite|improve this question

edited Mar 28 at 19:53

Bernard

124k741118

asked Mar 28 at 19:47

DanicaDanica

314

$endgroup$

add a comment | 

2

$begingroup$

Let $X$ be a cosimplicial space and suppose that the matching maps:
$$
s:X^krightarrow M^k-1X
$$
are fibrations for all $kleq n$. On one hand we can form the partial totalization $Tot_n(X)$, on the other hand we can take the fibrant replacement $RX$ and form $widetildeTot_n(X):=Tot_n(RX)$.

Do we have $Tot_n(X)simeq widetildeTot_n(X)$?

In other words, is the partial totalization in this case already homotopy invariant?

For example, totalization preserves weak equivalences between Reedy fibrant cosimplicial spaces (i.e. those for which all the matching maps are fibrations): if $X$ and $Y$ are (levelwise) weakly equivalent and both are Ready fibrant, then $Tot(X)simeq Tot(Y)$.

I guess I am asking if there is an analogous statement for the case of partial totalizations, and would it be as simple as I suggested - partial totalization preserves weak equivalences between truncated cosimplicial spaces if they are ''fibrant up to appropriate degree''.

algebraic-topology homotopy-theory simplicial-stuff monoidal-categories

share|cite|improve this question

edited Mar 28 at 19:53

Bernard

124k741118

asked Mar 28 at 19:47

DanicaDanica

314

$endgroup$

add a comment | 

$begingroup$

Let $X$ be a cosimplicial space and suppose that the matching maps:
$$
s:X^krightarrow M^k-1X
$$
are fibrations for all $kleq n$. On one hand we can form the partial totalization $Tot_n(X)$, on the other hand we can take the fibrant replacement $RX$ and form $widetildeTot_n(X):=Tot_n(RX)$.

Do we have $Tot_n(X)simeq widetildeTot_n(X)$?

In other words, is the partial totalization in this case already homotopy invariant?

For example, totalization preserves weak equivalences between Reedy fibrant cosimplicial spaces (i.e. those for which all the matching maps are fibrations): if $X$ and $Y$ are (levelwise) weakly equivalent and both are Ready fibrant, then $Tot(X)simeq Tot(Y)$.

I guess I am asking if there is an analogous statement for the case of partial totalizations, and would it be as simple as I suggested - partial totalization preserves weak equivalences between truncated cosimplicial spaces if they are ''fibrant up to appropriate degree''.

algebraic-topology homotopy-theory simplicial-stuff monoidal-categories

share|cite|improve this question

edited Mar 28 at 19:53

Bernard

124k741118

asked Mar 28 at 19:47

DanicaDanica

314

$endgroup$

share|cite|improve this question

edited Mar 28 at 19:53

Bernard

124k741118

asked Mar 28 at 19:47

DanicaDanica

314

share|cite|improve this question

edited Mar 28 at 19:53

Bernard

124k741118

asked Mar 28 at 19:47

DanicaDanica

314

edited Mar 28 at 19:53

Bernard

124k741118

edited Mar 28 at 19:53

Bernard

124k741118

asked Mar 28 at 19:47

DanicaDanica

314

asked Mar 28 at 19:47

DanicaDanica

314