### Background

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

notleft 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.]

`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