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• CommentRowNumber1.
• CommentAuthorZhen Lin
• CommentTimeAug 3rd 2013

I started a page about the simplicial bar construction. I haven’t checked all the details carefully (especially regarding $\mathcal{V}$-naturality!) though.

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeAug 3rd 2013

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.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeAug 3rd 2013
• (edited Aug 3rd 2013)

Or perhaps just call it ’realization’ (or ’realisation’ as the case may be).

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeAug 3rd 2013

@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.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeAug 3rd 2013

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.

• CommentRowNumber6.
• CommentAuthorTim_Porter
• CommentTimeAug 3rd 2013

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.

• CommentRowNumber7.
• CommentAuthorZhen Lin
• CommentTimeAug 3rd 2013

I replaced ‘geometric realisation’ with ‘realisation’ and fixed a little error in the statement of the hocolim theorem.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeAug 4th 2013

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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