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I've got a little confused by some set theory underlying Hopkins-Lurie's survey on classification of topological field theories.

The objects of $(\infty,n)$-categories need to form a set, at least if the definition we take is as some kind of weakened Kan complex. Yet in some examples they give objects clearly do not form sets, for example the $ (\infty,1)$-category of chain complexes $chain_t (k)$.

Is there something I am missing?

Edit: About comment below. I am not sure I understand what Lurie's 1.2.15 says concretely, about the example above. If someone could comment it would be great.

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    $\begingroup$ Isn't this addressed in Section 1.2.15 of HTT? Lurie is using universes. $\endgroup$ Sep 17, 2012 at 16:33
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    $\begingroup$ If you believe in categories with a proper class of objects, you also believe in simplicial sets with a proper class of vertices. I don't think any model for higher categories introduces any set-theoretic problems that aren't already there for 1-categories. $\endgroup$ Sep 18, 2012 at 15:26
  • $\begingroup$ I don't see how it is such a straight forward step, a simplicial set is functor $\Delta ^{op}} \to Set$ what do you replace this by? Granted it's probably my ignorance of set theory. But do proper classes form a category, with Set a subcategory? $\endgroup$
    – yasha
    Sep 18, 2012 at 16:43
  • $\begingroup$ I should have asked: do classes form a category? And if not, how do you define "simplicial classes". $\endgroup$
    – yasha
    Sep 18, 2012 at 18:45
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    $\begingroup$ I know what you meant. The collection of all morphisms forms a proper class anyway, and even if you only want to talk about the individual hom-sets, they are indexed by a proper class, so you need maps of proper classes to define a large category no matter how you look at it. $\endgroup$ Sep 20, 2012 at 16:05

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