nLab dimension of a cell complex

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Idea

The dimension of a cell complex XX is the largest natural number nn \in \mathbb{N}, if such exists, for which there are non-trivial nn-cells in XX. If there is no such largest nn the dimension is also said to be infinite.

Definition

Specifically, if XX is a CW-complex

Xlim nX n X \;\simeq\; \underset{\longrightarrow_{\mathrlap{n}}}{\lim} X_n

where each X nX_n is a pushout of the form

iI nS n1 X n1 (po) iI nD n X n \array{ \underset{ i \in I_n }{\sqcup} S^{n-1} &\longrightarrow& X_{n-1} \\ \big\downarrow &(po)& \big\downarrow \\ \underset{ i \in I_n }{\sqcup} D^{n} &\longrightarrow& X_n }

the dimension of XX is the largest nn for which the indexing sets I nI_n are non-empty.

notion of dimension

Last revised on March 21, 2021 at 07:30:24. See the history of this page for a list of all contributions to it.