Remark 6.3 (0MIJ) · ScalingStacks

[0MIJ]

Remark 6.3. We now examine the fiber products of cells in detail. We aim to express these fiber products as simple iterated colimits of cells. Let $\varphi:C_{i}\to C_{j}$ and $\psi:C_{k}\to C_{j}$ be a pair of functors ( $i,j,k\geq 0$ ). A map of cells $\varphi:C_{i}\to C_{j}$ either factors as a composite $C_{i}\to C_{0}\to C_{j}$ or is a suspension $\varphi=\sigma(\xi)$ of some map $\xi:C_{i-1}\to C_{j-1}$ .

We thus begin by contemplating the case in which $\varphi$ is not the suspension of a map of lower dimensional cells. In this case we have a diagram of pullback squares

Here $F$ is the fiber of $\psi:C_{k}\to C_{j}$ over the unique object in the image of $\varphi$ . There are four possibilities:

(A)

The image of $\psi$ may be disjoint from the image of $\varphi$ , in which case $F=\partial C_{0}=\emptyset$ . Hence $F$ and also $C_{i}\times F$ are the empty colimit of cells.
(B)

The fiber may be a zero cell, $F=C_{0}$ , in which case $C_{i}\times F\cong C_{i}$ is trivially a colimit of cells.
(C)

The fiber may be the $k$ -cell $F\cong C_{k}$ , but we have $i=0$ . In this case $C_{i}\times F\cong F\cong C_{k}$ is again trivially a colimit of cells.
(D)

The fiber may be an $k$ -cell $F\cong C_{k}$ , and we have $i\geq 1$ . In this case we have (cf. [34, Proposition 4.9])

$C_{i}\times C_{k}\cong(C_{i}\cup^{C_{0}}C_{k})\cup^{\sigma(C_{i-1}\times C_{k-1})}(C_{k}\cup^{C_{0}}C_{i})$

where for each pushout $C_{x}\cup^{C_{0}}C_{y}$ , the object $C_{0}$ is included into the final object of $C_{x}$ and the initial object of $C_{y}$ .

As the suspension functor $\sigma$ commutes with pullback squares, a general pullback of cells is the suspension of one of the types just considered. Moreover, as the suspension functor also commutes with pushout squares, the above considerations give a recipe for writing any fiber product of cells as an iterated pushout of cells. This will be made precise in Lemma 6.7.