It is fairly well known that if a full subcategory $i: C \hookrightarrow D$ has a left adjoint $F: D \to C$, then the canonical counit $F i(c) \to c$ is an isomorphism. (A classical example is the sheafification $a: \textbf{Set}^{C^{op}} \to Sh(C, J)$ for a Grothendieck site, left adjoint to the inclusion of sheaves into presheaves.) Thus left adjoints $F$ are "retractions" of full inclusions $i$, in the good categorical sense that $F i$ is, not equal necessarily, but naturally isomorphic to the identity functor $1_C$ on $C$, canonically so via the counit.

This applies in particular to the Yoneda embedding of a locally small category $y_C: C \to \textbf{Set}^{C^{op}}$, which is of course a full embedding. Having such a left adjoint is a strong cocompleteness condition on $C$; this notion of was introduced by Street and Walters and is called total cocompleteness. Most of the good categories of structures on sets arising in practice are totally cocomplete, or just total for short; these include categories of algebras of monads on $\textbf{Set}$, categories that are topological over $\textbf{Set}$, locally presentable categories, and more. The notion also makes sense in enriched category theory. A useful introductory treatment, paying due attention to foundational matters, is by Kelly.

Here's something I noticed recently that I found a little odd. Consider the context of preorders, which can be defined as categories enriched in the cartesian closed category $\mathbf{2} = \{0 \leq 1\}$. We have the converse statement:

If $P$ is a preorder (e.g., a poset), then $y_P: P \to \mathbf{2}^{P^{op}}$ has a retraction $s: \mathbf{2}^{P^{op}} \to P$ only if $s$ is left adjoint to $y_P$. In particular, due to uniqueness of left adjoints, $y_P$ can have at most one retraction (!).

Here $\mathbf{2}^{P^{op}}$ can be identified with the poset $\text{Down}(P)$ consisting of downward-closed subsets of $P$ (identify $\phi: P^{op} \to \mathbf{2}$ by the down-set $\phi^{-1}(1)$). In this case the Yoneda embedding is the principal down-set map, taking $p \in P$ to the principal down-set $\text{prin}(p) = \{q \in P: q \leq p\}$. Now suppose $s$ is a preorder map that retracts $\text{prin}: P \to \text{Down}(P)$; let $D$ be a down-set. For all $d \in D$ we have $\text{prin}(d) \subseteq D$, and so $d = s(\text{prin}(d)) \leq s(D)$ in $P$; this means $s(D)$ is an upper bound of $D$. On the other hand, for any upper bound $u$ of $D$, we have $D \subseteq \text{prin}(u)$, and therefore $s(D) \leq s(\text{prin}(u)) = u$, so $s(D)$ is a least upper bound of $D$. By uniqueness of least upper bounds, this means $s$ must be the supremum map $s = \text{sup}: \text{Down}(P) \to P$, making $P$ a sup-lattice (and indeed the supremum map on a sup-lattice $P$ is left adjoint to the Yoneda = principal embedding).

This example can be beefed up slightly, in various directions. If $P$ is any (let's say a small) category all of whose endomorphisms are identities (for example if $P$ is a preorder), then the Yoneda embedding $y= y_P: P \to \textbf{Set}^{P^{op}}$ again has at most one retraction pair $(s, \phi)$. For if we put $e = y s$, then $e\phi: e y = y s y \to y$ is an isomorphism and (using a little coend calculus) the evident composite map

$$F \cong \int^p F(p)\cdot y(p) \cong \int^p F(p) \cdot ey(p) \to eF$$

yields a natural unit $u F: F \to y s(F)$ for which the triangular equations for the putative adjunction $s \dashv y$ (with $\phi$ as counit) do in fact commute, by the endomorphism assumption on $P$ and fullness of $y$. In another direction, basically the same construction works in $V$-$\textbf{Cat}$, taking $V$ to be commutative quantale (which is equivalent to being a symmetric monoidal closed cocomplete small category, by a theorem of Freyd).

The Question

Define a retraction of a functor $i: C \to D$ to be a pair consisting of a functor $s: D \to C$ and an isomorphism $\phi: s i \stackrel{\sim}{\to} 1_C$. Is there an example of locally small $C$ whose Yoneda embedding $y_C$ admits a retraction $(s, \phi)$ but where $s$ is not left adjoint to $y_C$ with counit $\phi$?

[For those desiring some set-theoretic security: go ahead and interpret all categories as structures in a model $V$ of ZFC where $V$ has a strongly inaccessible cardinal $\kappa$, taking $\textbf{Set} = \text{Set}_\kappa$ to be the category of "small" sets (cardinality less than $\kappa$), and taking $C$ as locally small. You can assume $C$ is of "moderate size" ($Ob(C)$ has cardinality $\kappa$) if you like, or not -- your choice.]

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    $\begingroup$ Just a very quick comment. If idempotents split in C then I would hazard a guess that s must be left adjoint - else not. Theorem 3.5 of http://home.math.au.dk/kock/msau.PDF" and the discussion above it may be relevant, as may be the note Consequences of splitting idempotents" off Ross Street's webpage. $\endgroup$ – john Dec 29 '14 at 10:54
  • $\begingroup$ @john Thank you! I knew of Kock's paper, but that result I hadn't picked up before. That's helpful. $\endgroup$ – Todd Trimble Dec 29 '14 at 13:51
  • $\begingroup$ Just to have it recorded here, Theorem 3.5 of Kock's paper (and its proof) says that if there is a retraction, then there is a left adjoint obtained by splitting an idempotent of the retraction functor. So the question is whether that idempotent can be nontrivial. $\endgroup$ – Mike Shulman Dec 31 '14 at 6:02
  • $\begingroup$ In fact Prop 3 on Street's paper mentioned implies that idempotents split in $C$. Combining this with what Mike says should consequently ensure that $y_{c}$ always has a left adjoint, although perhaps you have to change $s$ itself? $\endgroup$ – john Jan 9 '15 at 12:25
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    $\begingroup$ @IvanDiLiberti Sorry for the delay. It's Anders Kock's paper on when multiplication structures are left adjoint to units (aka KZ monads): Journal of Pure and Applied Algebra 104 (1995), 41-59. $\endgroup$ – Todd Trimble Aug 29 at 19:12

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