I just want to ask if there is any deeper motivation or clear geometric "sense" behind the barycentric subdivision. Some friend asked me about this a few months ago, looking back the section at Hatcher, I still feel quite confused. I remember one friend told me combinatorically one can do this from posets back to posets, but this does not give me any way to "understand" it properly. In some books (Bredon, for example), the author use excision property as one of the axioms, I'm wondering "where they came from, why they make any sense?".

More precisely, one may also raise the question that since barycentric subdivision made it possible to calculate homology via smaller and smaller subdivisions, since if one "divide" infinitely, one will get to something resembling fractals. My friend told me that since fractals do not endow a nice simplicial structure, it has no definite homology groups. But can we overcome this by using infinite many times of barycentric subdivisions? I don't know much about analysis on fractals, so I ask in here.

I just want to ask if there is any deeper motivation or clear geometric "sense" behind the barycentric subdivision. Some friend asked me about this a few months ago, looking back the section at Hatcher, I still feel quite confused. I remember one friend told me combinatorically one can do this from posets back to posets, but this does not give me any way to "understand" it properly. In some books (Bredon, for example), the author use excision property as one of the axioms, I'm wondering "where they came from, why they make any sense?".