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Let $C$ be a locally small category. Then we have the Yoneda embedding $Y : C \to \widehat{C} := [C^{op},Set]$. Recall that $C$ is called total when $Y$ has a left adjoint.

My first question is: Why aren't there any set-theoretic obstructions for such a left adjoint to exist? The category $\widehat{C}$ is not locally small, lives in a bigger universe than $C$. In some papers I've read that, for instance, algebraic categories are total. Ok, but how does the left adjoint of $Y$ looks like explicitly for $C=\mathsf{Ring}$ for instance? I'm not yet convinced that such a functor can exist ...

The second question is: What do you think of the following variant of totality (and is it already known and studied in the literature): Define $\widehat{C}$ to be the category of presheaves on $C$ which are small colimits of representable functors. Equivalently, these presheaves are cofinally small. Then $\widehat{C}$ lives in the same universe of $C$, and in fact the Yoneda embedding enjoys the universal property of the universal cococompletion of $C$ (see here).

In this setting, I can show that the category of affine schemes $\mathsf{Aff}$ is total (i.e. $\mathsf{Ring}$ is cototal), and in fact the corresponding adjunction is quite well-known: To every affine scheme $X$ we associate its functor of points $\hom(-,X)$. To every cofinally small presheaf on affine schemes $F$ we associate $\mathcal{O}(F) := \hom(F,\mathbb{A}^1)$ with component-wise ring operations (or rather we associate $\mathrm{Spec} \mathcal{O}(F)$). Note that we could also do this construction for every presheaf $F$ (and this is done in many texts, for example in J. S. Milne's script on algebraic groups), but then $\mathcal{O}(F)$ has no chance of being a small set and therefore an object of $\mathsf{Ring}$.

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I've had similar thoughts in reading mathoverflow.net/questions/83149/… –  Buschi Sergio Jun 5 '12 at 13:10
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Totality is for example a consequence of cocompleteness and the existence of a small dense subcategory. If you have such a subcategory, with inclusion $K\colon A \rightarrow C$, say, then the functor $\widetilde{K} \colon C \rightarrow [A^{\mathrm{op}},\mathrm{Set}]$ which sends $c \in C$ to $C(K-,c)$ is fully faithful. Cocompleteness of $C$ implies that it has a left adjoint $L$. The reflection of a presheaf $F \colon C^{\mathrm{op}} \rightarrow \mathrm{Set}$ is then given by $L(F \circ K^{\mathrm{op}})$. To see this, note that for every $c \in C$, the functor $C(-,c)$ is the right Kan extension of $C(K-,c)$ along $K^{\mathrm{op}}$, which in turn follows from the fact that $C(-,c)$ is continuous and $A$ is dense. See for example

A survey of totality for enriched and ordinary categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 2 (1986), p. 109-132

(Section 6) for more examples and a more elaborate theorem, of which the above is a consequence.

Your second suggestion seems to me to be equivalent to cocompleteness of the category in question. Tautologically, every representable functor admits a reflection, given by the representing object. If your category is (small) cocomplete, then the presheaves which admit reflections are closed under small colimits.

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Daniel, I am a bit suspicious about your claim that Martin's condition is equivalent to cocompleteness. Could you elaborate more on it? –  Michal R. Przybylek Jun 6 '12 at 8:11
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A fancier way to look at this: the category of small colimits of representables is the free cocompletion of a locally small category. It forms the object part of a pseudomonad of the Kock-Zoeberlein (or lax-idempotent) type on the category of locally small categories. Thus a locally small category is (small) cocomplete if and only if the unit (Yoneda embedding) has a left adjoint. See e.g. Section 2 of "Lex colimits" by Garner and Lack, Propositions 2.1 and 2.2. –  Daniel Schäppi Jun 6 '12 at 10:02
    
(here by "Yoneda embedding" I mean the corestriction of the usual Yoneda embedding to small presheaves) –  Daniel Schäppi Jun 6 '12 at 10:05
    
Thx, Daniel :-) –  Michal R. Przybylek Jun 6 '12 at 10:56
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