# Classifying space of large category?

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?

• Well, usually people like their topological spaces to have a set of points, or their simplicial sets to be made of, well, sets. – Mariano Suárez-Álvarez Mar 28 '11 at 1:48
• 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. – Martin Brandenburg Mar 28 '11 at 2:14

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.
• $C_0$ and $C_1$ are subcategories of which category? – David Roberts Apr 6 '11 at 5:57
• Both categories are in $\mathcal{C}$. – Spice the Bird Apr 7 '11 at 17:58
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.