Dimension's in pooling layers

why don’t we the concept of convolution over volume and get the Nc equals to 1 instead of Nc=Nc

Not sure I understand the question, but convolution and pooling are two different things. The definition of pooling layers is that they operate “channelwise”, so that nC_{out} = nC_{in} by definition. In a convolution layer, the number of filters is an independent choice, so that there is no connection between nC_{in} and nC_{out}.

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