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Is there a standard construction of a classifying space $BC$ for a category $C$ which is enriched which takes into account the enrichment?

This is of course vague... The simplest example I can think of arises when $C$ is enriched over abelian groups or $k$-modules for some commutative ring $k$.

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1 Answer 1

For a category enriched in topological spaces, the usual classifying space can be made to take into account the topology on morphisms. More generally this works for categories internal to top and is described here Segal "Classifying spaces and spectral sequences". Even more generally if you have a category enriched in a category C, and C has enough fiber products, then the nerve makes sense as a simplicial-C object. You can then try to construct the classifying space using a coend and a suitable cosimplicial object $\Delta \to C$ (In the case $C = Top$ this is the usual embedding).

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I've found Segal's paper a bit frustrating at times. When he talks about topological categories he asks that the four "obvious" structure maps be continuous. It took some guessing on my part to unpack his meaning. Do you have another reference, which you like, that outlines these various definitions, constructions and such? –  Justin Curry Mar 9 '10 at 3:29
Do you have any references where the 'even more general' case is dealt with or used? The topological case is the one I knew about, and my question was really intended to include the other case (I should of course have said so!) –  Mariano Suárez-Alvarez Mar 13 '10 at 15:09

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