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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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2 Answers 2

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

Edit: Modified in accordance with Tom Leinster's entirely reasonable objections.

Sorry to exhume this question from 5+ years ago. In case someone is still looking for an answer, note that a very specific version (restricting to $V = \text{Cat}$) is addressed up to homotopy in the paper

M Bullejos and A Cegarra, On the geometry of 2-categories and their classifying spaces. K-theory, 29:211 – 229, (2003).

using the Duskin/Street nerve construction. Given a $V$-enriched category $C$, one constructs the simplicial set $\Delta C$ as follows: vertices are the objects of $C$, and higher simplices spanning objects $x_0,\cdots, x_d$ consist of

  1. 1-morphisms $f_{ij}:x_i \to x_j$ for $0 \leq i \leq j \leq d$, and
  2. 2-morphisms $\alpha_{ijk}:f_{ik} \Rightarrow f_{jk} \circ f_{ij}$ for $0 \leq i \leq j \leq k \leq d$ subject to certain associativity and identity constructions (see the Introduction of the linked paper for details).

One can also construct the Segal nerve as outlined in Chris Schommer-Pries's answer, but it is not as directly related to the objects and morphisms of the underlying enriched category. Bullejos and Cegarra show in the linked paper that the two constructions are naturally homotopy-equivalent so it doesn't matter too much either way!

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I think that it's still 2015 :-) –  Peter Samuelson Aug 27 at 1:14
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@PeterSamuelson Thanks, now fixed :) –  Vidit Nanda Aug 27 at 19:04
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Vidit, I'm not sure what you mean. The 0-simplices of $\Delta C$ are the objects of $C$, sure. But if $x_0$ and $x_1$ are objects of $C$, what is the set of 1-simplices in $\Delta C$ from $x_0$ to $x_1$? And in your item 1, is $f_{ij}$ just any object of $V$? Finally, is this really answered in Bullejos and Cegarra's paper? I could only see the special case $V = \mathbf{Cat}$, i.e. classifying spaces of 2-categories (as their title suggests). –  Tom Leinster Aug 28 at 2:11
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@TomLeinster Hmm, you're right of course, I'm not sure what I mean any more either. I'm editing to clarify that this only works for $Cat$-enrichment. –  Vidit Nanda Aug 29 at 3:11
    
Thanks for the clarification. –  Tom Leinster Aug 29 at 15:54

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