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I started a page about the simplicial bar construction. I haven’t checked all the details carefully (especially regarding $\mathcal{V}$-naturality!) though.
Just one point, is ’geometric realization’ really a good term for something where there is no necessity for the resulting objects to have any geometric nature. Perhaps some term such as $\mathcal{M}$-realization with a note that in cases such as …. this is the geometric realization.
Or perhaps just call it ’realization’ (or ’realisation’ as the case may be).
@Tim: Maybe not, but isn’t it pretty well established? Even the classical case of the geometric realization of a simplicial set as a topological space doesn’t have any “geometry” in the sense I usually think of the word, only a topology.
Really, geometric realization is the established term even when the receiving category isn’t a good category of (topological) spaces?
We do have this page nerve and realization, to which we can point for more general situations.
The term realisation is used for this on the page , nerve and realization, so perhaps should be used here. The general situation is very well handled there.
I replaced ‘geometric realisation’ with ‘realisation’ and fixed a little error in the statement of the hocolim theorem.
I’ve always heard it called “geometric realization” in at least some other situations, like spectra. But it doesn’t matter.
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