is the presheaf category of a locally small category locally small? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:58:14Z http://mathoverflow.net/feeds/question/24540 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24540/is-the-presheaf-category-of-a-locally-small-category-locally-small is the presheaf category of a locally small category locally small? apk 2010-05-13T20:52:34Z 2010-05-13T22:52:15Z <p>e.g. is $\widehat{\mathbf{SET}}$ locally small?</p> http://mathoverflow.net/questions/24540/is-the-presheaf-category-of-a-locally-small-category-locally-small/24549#24549 Answer by Todd Trimble for is the presheaf category of a locally small category locally small? Todd Trimble 2010-05-13T22:37:45Z 2010-05-13T22:37:45Z <p>There's a theorem due to Freyd and Street that says that if $C$ is locally small and the presheaf category $Set^{C^{op}}$ is locally small, then $C$ is equivalent to a small category. See <a href="http://www.tac.mta.ca/tac/volumes/1995/n9/v1n9.pdf" rel="nofollow">this paper</a>. </p> http://mathoverflow.net/questions/24540/is-the-presheaf-category-of-a-locally-small-category-locally-small/24551#24551 Answer by Andrej Bauer for is the presheaf category of a locally small category locally small? Andrej Bauer 2010-05-13T22:52:15Z 2010-05-13T22:52:15Z <p>No, not in general. Todd gave a useful general answer and here is a concrete counter-example.</p> <p>Let $\mathbf{C}$ be a category whose objects are sets and where the only morphisms are the identities. Consider the functors $F, G : \mathbf{C} \to \mathbf{Set}$ given by $F(X) = 1 = \lbrace 0 \rbrace$ and $G(X) = 2 = \lbrace 0,1\rbrace$. Because $\mathbf{C}$ is discrete, every family of maps $(\eta_X : F(X) \to G(X))_{X \in \mathbf{C}}$ is natural. Therefore, natural transformations $F \to G$ are in bijective correspondence with classes:</p> <ul> <li><p>a natural transformation $\eta : F \to G$ corresponds to the class $\lbrace X \in \mathbf{Set} \mid \eta_X(0) = 1\rbrace$.</p></li> <li><p>a class $C$ corresponds to the natural transformation $\eta : F \to G$ given by</p> <blockquote> <p>$\eta_X(0) = 1$ if $X \in C$ and $\eta_X(0) = 0$ if $X \not\in C$.</p> </blockquote></li> </ul> <p>Thus the presheaf category $\widehat{\mathbf{C}}$ is not locally small (note that $\mathbf{C}^{op} = \mathbf{C}$).</p> <p>I cannot think at the moment of concrete presheaves $F$ and $G$ on $\mathbf{Set}$ for which the natural transformations $F \to G$ form a proper class. Someone please help.</p>