- Let $\catl{C}$ be a small category. Denote by $\word{Cont}(\catl{C}^{op}, \catw{Set})$ the full subcategory of presheaves $\catw{Set}^{\catl{C}^{op}}$ that consists of such $\mor{H}{\catl{C}^{op}}{\catw{Set}}$ that preserve small limits that existsexist in $\catl{C}$$\catl{C}^{op}$ (i.e. map existing colimits from $\catl{C}$ to limits in $\catw{Set}$). It follows from abstract nonsense that $\word{Cont}(\catl{C}^{op}, \catw{Set})$ has small (co)limits and, in fact, is a reflective subcategory of $\catw{Set}^{\catl{C}^{op}}$.
- Let $\catl{C}$ be a small category. Denote by $\word{Cont}(\catl{C}^{op}, \catw{Set})$ the full subcategory of presheaves $\catw{Set}^{\catl{C}^{op}}$ that consists of such $\mor{H}{\catl{C}^{op}}{\catw{Set}}$ that preserve small limits that exists in $\catl{C}$ (i.e. map colimits from $\catl{C}$ to limits in $\catw{Set}$). It follows from abstract nonsense that $\word{Cont}(\catl{C}^{op}, \catw{Set})$ has small (co)limits and, in fact, is a reflective subcategory of $\catw{Set}^{\catl{C}^{op}}$.
- Let $\catl{C}$ be a small category. Denote by $\word{Cont}(\catl{C}^{op}, \catw{Set})$ the full subcategory of presheaves $\catw{Set}^{\catl{C}^{op}}$ that consists of such $\mor{H}{\catl{C}^{op}}{\catw{Set}}$ that preserve small limits that exist in $\catl{C}^{op}$ (i.e. map existing colimits from $\catl{C}$ to limits in $\catw{Set}$). It follows from abstract nonsense that $\word{Cont}(\catl{C}^{op}, \catw{Set})$ has small (co)limits and, in fact, is a reflective subcategory of $\catw{Set}^{\catl{C}^{op}}$.
- Let $\catl{C}$ be a small category. Denote by $\word{Cont}(\catl{C}^{op}, \catw{Set})$ the full subcategory of presheaves $\catw{Set}^{\catl{C}^{op}}$ that consists of such $\mor{H}{\catl{C}^{op}}{\catw{Set}}$ that preserve small limits that exists in $\catl{C}$ (i.e. map colimits from $\catl{C}$ to limits in $\catw{Set}$). It follows from abstract nonsense that $\word{Cont}(\catl{C}^{op}, \catw{Set})$ has small (co)limits and, in fact, is a reflective subcategory of $\catw{Set}^{\catl{C}^{op}}$.
Because $1$ is costrict in $\catl{C}$ we have that $1 \sqcup 1 \approx 1$ in $\catl{C}$ and since coproduct and the terminal object are preserved by $E$, we have also $1 \sqcup 1 \approx 1$ in $\overline{\catl{C}}$. Let $\mor{f,g}{1}{A}$ be two morphisms in $\overline{\catl{C}}$. By the universal property of coproducts they induce the copairing morphism $\mor{[f,g]}{1 \sqcup 1}{A}$ that commutes with the coproduct's injections. However, since $1\approx 1 \sqcup 1$, the coproducts injections are identities and so $f = [f,g] = g$. Therefore, for every $A \in \overline{\catl{C}}$ there is at most one arrow $1 \rightarrow A$. If we take for $A$ an exponent $E(Y)^{E(X)}$, then by cartesian clossedness of $\overline{\catl{C}}$ and fully faithfulness of $E$:
$$\hom_\catl{C}(X, Y) \approx \hom_\overline{\catl{C}}(E(X), E(Y)) \approx \hom_\overline{\catl{C}}(1, E(Y)^{E(X)}) \leq 1$$$$\hom_\catl{C}(X, Y) \leq \hom_\overline{\catl{C}}(E(X), E(Y)) \approx \hom_\overline{\catl{C}}(1, E(Y)^{E(X)}) \leq 1$$
Therefore (by the second property) $\overline{\mathcal{Z}_3}$ is the smallest reflective subcategory of $\catw{Set}^{\mathcal{Z}_3^{op}}$ containing representables.
- The Dedekind-MacNeille construction as described in Todd's answer works when the monad induced by the Isbell conjugation is itself idempotent. For example, this is always true in the world of posets, in the world of Lawvere metric spaces, and more generally in the world of categories enriched over affine quantales (because in such a case every enriched monad is idempotent).
I actually think that only Dedekind-MacNeille completitions for categories enriched over (co)complete posets deserve the name.
The reason is that there is a subtlety here (which may look minor at first, but I think is crucial for the whole construction) that makes that the direct categorification of the original Dedekind-MacNeille requirements for completion have no sense --- the completion should be complete and cocomplete, which means that it should be closed under all limits/colimits. On the other hand, the term "a category is complete and cocomplete" means that the category is closed under small limits and small colimits (obviously, there is no (co)completion under all (co)limits in a $\catw{Set}$-enriched world). Therefore, the direct categorification of the requirement is possible only when the base of the enrichment has all limits/colimits, which in turn, implies that the base of the enrichment is a poset (at least in the classical mathematics).
As I said, the distinction between small and all may look minor at first, but in fact the distinction is not about quantity, but about quality. This distinction shows up on many occasions. For example, constructively if a category is complete (with respect to all cones) then it is also cocomplete (with respect to all cocones), and every colimit can be expressed as a limit; however, this is no longer true if the category is only small complete --- the cone from the canonical representation of a small cocone may be large, and there may be no reason for its limit to exists.
- Let $\catl{C}$ be a small category. Denote by $\word{Cont}(\catl{C}^{op}, \catw{Set})$ the full subcategory of presheaves $\catw{Set}^{\catl{C}^{op}}$ that consists of such $\mor{H}{\catl{C}^{op}}{\catw{Set}}$ that preserve small limits (i.e. map colimits from $\catl{C}$ to limits in $\catw{Set}$). It follows from abstract nonsense that $\word{Cont}(\catl{C}^{op}, \catw{Set})$ has small (co)limits and, in fact, is a reflective subcategory of $\catw{Set}^{\catl{C}^{op}}$.
Because $1$ is costrict in $\catl{C}$ we have that $1 \sqcup 1 \approx 1$ in $\catl{C}$ and since coproduct and the terminal object are preserved by $E$, we have also $1 \sqcup 1 \approx 1$ in $\overline{\catl{C}}$. Let $\mor{f,g}{1}{A}$ be two morphisms in $\overline{\catl{C}}$. By the universal property of coproducts they induce the copairing morphism $\mor{[f,g]}{1 \sqcup 1}{A}$ that commutes with the coproduct's injections. However, since $1\approx 1 \sqcup 1$, the coproducts injections are identities and so $f = [f,g] = g$. Therefore, for every $A \in \overline{\catl{C}}$ there is at most one arrow $1 \rightarrow A$. If we take for $A$ an exponent $E(Y)^{E(X)}$, then by cartesian clossedness of $\overline{\catl{C}}$ and fully faithfulness of $E$:
$$\hom_\catl{C}(X, Y) \approx \hom_\overline{\catl{C}}(E(X), E(Y)) \approx \hom_\overline{\catl{C}}(1, E(Y)^{E(X)}) \leq 1$$
Therefore (by the second property) $\overline{\mathcal{Z}_3}$ is the smallest reflective subcategory of $\catw{Set}^{\mathcal{Z}_3^{op}}$ containing representables.
- Let $\catl{C}$ be a small category. Denote by $\word{Cont}(\catl{C}^{op}, \catw{Set})$ the full subcategory of presheaves $\catw{Set}^{\catl{C}^{op}}$ that consists of such $\mor{H}{\catl{C}^{op}}{\catw{Set}}$ that preserve small limits that exists in $\catl{C}$ (i.e. map colimits from $\catl{C}$ to limits in $\catw{Set}$). It follows from abstract nonsense that $\word{Cont}(\catl{C}^{op}, \catw{Set})$ has small (co)limits and, in fact, is a reflective subcategory of $\catw{Set}^{\catl{C}^{op}}$.
Because $1$ is costrict in $\catl{C}$ we have that $1 \sqcup 1 \approx 1$ in $\catl{C}$ and since coproduct and the terminal object are preserved by $E$, we have also $1 \sqcup 1 \approx 1$ in $\overline{\catl{C}}$. Let $\mor{f,g}{1}{A}$ be two morphisms in $\overline{\catl{C}}$. By the universal property of coproducts they induce the copairing morphism $\mor{[f,g]}{1 \sqcup 1}{A}$ that commutes with the coproduct's injections. However, since $1\approx 1 \sqcup 1$, the coproducts injections are identities and so $f = [f,g] = g$. Therefore, for every $A \in \overline{\catl{C}}$ there is at most one arrow $1 \rightarrow A$. If we take for $A$ an exponent $E(Y)^{E(X)}$, then by cartesian clossedness of $\overline{\catl{C}}$ and faithfulness of $E$:
$$\hom_\catl{C}(X, Y) \leq \hom_\overline{\catl{C}}(E(X), E(Y)) \approx \hom_\overline{\catl{C}}(1, E(Y)^{E(X)}) \leq 1$$
Therefore (by the second property) $\overline{\mathcal{Z}_3}$ is the smallest reflective subcategory of $\catw{Set}^{\mathcal{Z}_3^{op}}$ containing representables.
- The Dedekind-MacNeille construction as described in Todd's answer works when the monad induced by the Isbell conjugation is itself idempotent. For example, this is always true in the world of posets, in the world of Lawvere metric spaces, and more generally in the world of categories enriched over affine quantales (because in such a case every enriched monad is idempotent).
I actually think that only Dedekind-MacNeille completitions for categories enriched over (co)complete posets deserve the name.
The reason is that there is a subtlety here (which may look minor at first, but I think is crucial for the whole construction) that makes that the direct categorification of the original Dedekind-MacNeille requirements for completion have no sense --- the completion should be complete and cocomplete, which means that it should be closed under all limits/colimits. On the other hand, the term "a category is complete and cocomplete" means that the category is closed under small limits and small colimits (obviously, there is no (co)completion under all (co)limits in a $\catw{Set}$-enriched world). Therefore, the direct categorification of the requirement is possible only when the base of the enrichment has all limits/colimits, which in turn, implies that the base of the enrichment is a poset (at least in the classical mathematics).
As I said, the distinction between small and all may look minor at first, but in fact the distinction is not about quantity, but about quality. This distinction shows up on many occasions. For example, constructively if a category is complete (with respect to all cones) then it is also cocomplete (with respect to all cocones), and every colimit can be expressed as a limit; however, this is no longer true if the category is only small complete --- the cone from the canonical representation of a small cocone may be large, and there may be no reason for its limit to exists.
The following is what I wrote about six months ago during a private discussion (posted on request :-)
Let me start with a positive (obvious :-) result. $\newcommand{\word}[1] { \mathit{#1} } \newcommand{\catl}[1] { \mathbb{#1} } \newcommand{\catw}[1] { \mathbf{#1} } \newcommand{\mor}[3] { {#1 \colon #2 \rightarrow #3} }$
- Let $\catl{C}$ be a small category. Denote by $\word{Cont}(\catl{C}^{op}, \catw{Set})$ the full subcategory of presheaves $\catw{Set}^{\catl{C}^{op}}$ that consists of such $\mor{H}{\catl{C}^{op}}{\catw{Set}}$ that preserve small limits (i.e. map colimits from $\catl{C}$ to limits in $\catw{Set}$). It follows from abstract nonsense that $\word{Cont}(\catl{C}^{op}, \catw{Set})$ has small (co)limits and, in fact, is a reflective subcategory of $\catw{Set}^{\catl{C}^{op}}$.
Of course, every presheaf preserves colimits, therefore the restricted Yoneda embedding $A \mapsto \hom(-, A)$ gives functor: $$\mor{y}{\catl{C}}{\word{Cont}(\catl{C}^{op}, \catw{Set})}$$ This functor almost by definition preserves all limits that exist in $\catl{C}$. Observe, that it also preserves colimits. Let $\mor{F}{\catl{J}}{\catl{C}}$ be a functor from a small category, and assume that the colimit $\mathit{colim}(F)$ exists in $\catl{C}$. Consider any $H \in \word{Cont}(\catl{C}^{op}, \catw{Set})$. Then morphisms: $$y(\mathit{colim}(F)) = \hom(-, \mathit{colim}(F)) \rightarrow H(-)$$ by Yoneda, are tantamount to elements: $$H(\mathit{colim}(F)) \approx \mathit{lim}(H \circ F)$$ Similarly, morphisms: $$y(F(J)) = \hom(-, F(J)) \rightarrow H(-)$$ are tantamount to elements: $$H(F(J))$$ and one may easily verify, that the above exhibits $\hom(-, \mathit{colim}(F))$ as the limit $\mathit{lim}(y \circ F)$.
The above construction is described, for example, in "Basic concepts of enriched categories" of Max Kelly (Section 3.12). Moreover, as pointed out there, $\word{Cont}(\catl{C}^{op}, \catw{Set})$ inherits any monoidal (closed) structure from $\catl{C}$ via (restricted) convolution.
On the other hand, if $\catl{C}$ is not (monoidal) closed, then $\word{Cont}(\catl{C}^{op}, \catw{Set})$ generally will not be (monoidal) closed, due to the following fact.
- There is no universal limit and colimit preserving fully faithful embedding of a small category to a cartesian closed complete and cocomplete category.
In fact, there is no universal embedding into cartesian closed category that preserves terminal object and binary coproducts.
For let us assume that $\catl{C}$ is non-degenerated and has a costrict terminal object (terminal object $1$ is costrict if whenever there exists a morphism $1 \rightarrow X$ then $X \approx 1$) and binary coproducts (you may take $\catl{C} = \catw{FinSet}^{op}$), and there is such an embedding $\mor{E}{\catl{C}}{\overline{\catl{C}}}$.
Because $1$ is costrict in $\catl{C}$ we have that $1 \sqcup 1 \approx 1$ in $\catl{C}$ and since coproduct and the terminal object are preserved by $E$, we have also $1 \sqcup 1 \approx 1$ in $\overline{\catl{C}}$. Let $\mor{f,g}{1}{A}$ be two morphisms in $\overline{\catl{C}}$. By the universal property of coproducts they induce the copairing morphism $\mor{[f,g]}{1 \sqcup 1}{A}$ that commutes with the coproduct's injections. However, since $1\approx 1 \sqcup 1$, the coproducts injections are identities and so $f = [f,g] = g$. Therefore, for every $A \in \overline{\catl{C}}$ there is at most one arrow $1 \rightarrow A$. If we take for $A$ an exponent $E(Y)^{E(X)}$, then by cartesian clossedness of $\overline{\catl{C}}$ and fully faithfulness of $E$:
$$\hom_\catl{C}(X, Y) \approx \hom_\overline{\catl{C}}(E(X), E(Y)) \approx \hom_\overline{\catl{C}}(1, E(Y)^{E(X)}) \leq 1$$
which contradicts the fact that $\catl{C}$ is non-degenerated.
- Generally, the completions obtained in any of the mentioned ways will not be the smallest completion. In fact, for a general $\catl{C}$, the smallest reflective subcategory of $\catw{Set}^{\catl{C}^{op}}$ containing representables may not be a smallest limit and colimit completion of $\catl{C}$.
Let us take for example $\mathcal{Z}_3$ group thought as of a category with a single object. Actually one may easily compute the category $\overline{\mathcal{Z}_3}$ induced by the idempotent monad associated to the monad $T$ on the Isbell cojugation for $\mathcal{Z}_3$ (by the theorem of Fakir this category can be computed by taking the objects that respects $T$-weak equivalences). Explicitly, category $\overline{\mathcal{Z}_3}$ is the full subcategory of $\catw{Set}^{\mathcal{Z}_3^{op}}$ on free permutations, together with the terminal permutation (up to the terminal permutation it is equivalent to the Kleisli resolution of the monad $\mor{(-) \times \mathcal{Z}_3}{\catw{Set}}{\catw{Set}}$, but I'm not sure if this is deep or meaningless...).
One may easily see that $\overline{\mathcal{Z}_3}$ is not self-dual, thus cannot be the smallest limit and colimit completion (since $\mathcal{Z}_3$ is self-dual, if $\overline{\mathcal{Z}_3}$ was the smallest, then $(\overline{\mathcal{Z}_3})^{op}$ would be the smallest). Moreover, $\overline{\mathcal{Z}_3}$ satisfies the following properties:
- its every object is a colimit over $\mathcal{Z}_3$,
- its every object is a double-limit over $\mathcal{Z}_3$
Therefore (by the second property) $\overline{\mathcal{Z}_3}$ is the smallest reflective subcategory of $\catw{Set}^{\mathcal{Z}_3^{op}}$ containing representables.