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Whenever I've seen the definition of the classifying space of a category, the category is always specified to be small. I understand the definition well enough for my purposes (I think), but it occurred to me today, why small?

Is there a reason we only take nerves of small categories? Does the definition fail if the objects are too big?

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    $\begingroup$ Well, usually people like their topological spaces to have a set of points, or their simplicial sets to be made of, well, sets. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Mar 28, 2011 at 1:48
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    $\begingroup$ As usual you could just work with universes and accept that the classifying space of a $\mathcal{U}$-category is just a $\mathcal{U}^+$-small space. $\endgroup$
    – Martin Brandenburg
    Commented Mar 28, 2011 at 2:14

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Their is a notion of a category that is homotopicaly small. These categories have a nerve that is well defined up to homotopy. The notion is as follows: Let $\mathcal{C}$ be a large category. It is homotopicaly small if their is a subcategory, $\mathcal{D}$ such that for any small subcategory, $\mathcal{C}_0$ their is another subcategory, $\mathcal{C}_1$, with $\mathcal{C}_0\subset\mathcal{C}_1$, such that $\mathcal{D}$ id equivalent to $\mathcal{C}_1$. Thus one thinks of the nerve of $\mathcal{D}$ as the homotopy type of the category.

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  • $\begingroup$ $C_0$ and $C_1$ are subcategories of which category? $\endgroup$
    – David Roberts ♦
    Commented Apr 6, 2011 at 5:57
  • $\begingroup$ Both categories are in $\mathcal{C}$. $\endgroup$
    – Spice the Bird
    Commented Apr 7, 2011 at 17:58
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I will ignore the distinction here between the classifying space as a topological space or as a simplicial set. Fix for once and for all the adjunction $sSet \leftrightarrows Top$. Then giving the classifying topological space is 'just as good' as giving the classifying simplicial set. The 'classifying space' of a category qua simplicial set is then its nerve.

Consider the example of the category $Set$, which is a large category whatever foundational workaround you use. The nerve of $Set$ is not a functor $\Delta \to Set$, so one is forced (no pun intended) to use some sort of formalism that allows you to 'jump up a universe'. One could take NBG set theory, with classes and sets, and the nerve of $Set$ will be a simplicial class. Or one could take algebraic set theory (which is a categorical axiomatic formulation of sets and classes), or resort to Grothendieck universes or models arising from inaccessible cardinals, as Martin says in the comments.

I suppose one could try to work with simplicial classes informally, as a collection of sentences in first order logic, and the (explicitly given) maps between them, but I'm not sure how you can talk about class functions at this level. The only hope is that the face and degeneracy maps of the nerve of a category are very easy to describe, but I'm not an expert in that area.

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