Let $\mathcal{C}$ be any (perhaps small) category. Is there a (co)complete category $\mathcal{C}'$ and an inclusion $\mathcal{C}\hookrightarrow\mathcal{C}'$ which is universal among (co)complete categories containing $\mathcal{C}$? Perhaps something akin to the Yoneda embedding into the category of presheaves over $\mathcal{C}$ is the obvious choice? If not perhaps some other explanation can be given.
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1$\begingroup$ Justin, there is a very similar question and discussion taking place at mathoverflow.net/questions/55859/… You might also look at the nLab article: ncatlab.org/nlab/show/free+cocompletion Your surmise is basically correct. $\endgroup$– Todd TrimbleCommented Feb 19, 2011 at 2:25
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1$\begingroup$ See Kashiwara-Schapira, Categories and Sheaves. This question is discussed in first chapters. $\endgroup$– SashaCommented Feb 19, 2011 at 5:01
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The Yoneda $\mathcal{C}\to Psh(\mathcal{C})$ is initial among pairs (D,F) consisting of a cocomplete category $D$ and a functor $F:\mathcal{C}\to D$. That is, it's the initial object of the comma category $\mathcal{C}\downarrow_{Cat} CoComp$, where $CoComp$ is the category of cocomplete categories with colimit preserving functors between them (technically to form this comma category, we would want to form the comma category of the inclusions of both $\mathcal{C}$ and $\mathcal{CoComp}$ into $Cat$, but, hey, what's an abuse of notation between friends?).
For this reason, we call the category $Psh(\mathcal{C})$ the free cocompletion of $\mathcal{C}$
answered Feb 19, 2011 at 2:27
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$\begingroup$ I should note that the notion of smallness doesn't play any role in the definition. Smallness only affects whether or not the category $Psh(\mathcal{C})$ exists. $\endgroup$ Commented Feb 19, 2011 at 2:36
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5$\begingroup$ You may or may not believe that Psh(C) exists for large C. But it will not be the free cocompletion unless C is small. The free cocompletion is the category of small presheaves on C. A presheaf is small if it is a (small) colimit of representables; equivalently, if it is the left Kan extension of its restriction to some small full subcategory. $\endgroup$ Commented Feb 19, 2011 at 3:17
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$\begingroup$ @Steve: Yes, I was assuming that it didn't exist so I didn't have to talk about the situation where we have an annoying relative size difference. $\endgroup$ Commented Feb 19, 2011 at 4:06