The concept of Cauchy completeness, ordinarily thought of as applying to metric spaces, was vastly generalized by Bill Lawvere in his influential paper Metric spaces, generalized logic, and closed categories. It is now seen by category theorists as a concept of enriched category theory, with close ties to the concept of Morita equivalence in the theory of modules. In category theory one also speaks of idempotent completeness.
The basic idea is that the Cauchy completion of a category is the closure of a category under what are called “absolute limits”, i.e., those limits that are preserved by any functor whatsoever. Equivalently, the Cauchy completion is the closure with respect to absolute colimits. If $C$ is small, the Cauchy completion $\bar{C}$ of $C$ lies between $C$ and its “free cocompletion”, aka presheaf category
and consists of the presheaves $F$ dubbed tiny by Lawvere, meaning those presheaves which are connected and projective: the functor
preserves small coproducts and coequalizers. All of these concepts generalize straightforwardly to the context of general $V$-enriched categories, where $V$ is a complete, cocomplete, symmetric monoidal closed category.
Lawvere defines a point of the Cauchy completion of a small $V$-category $C$ to be a $V$-enriched bimodule $p: \mathbf{1} \to C$ (in other words, a $V$-functor $\mathbf{1} \to V^{C^{op}}$) for which there is a bimodule $q: C \to \mathbf{1}$ right adjoint to $p$ (in the bicategory of enriched bimodules, see profunctor), where $\mathbf{1}$ is the unit $V$-category. Thus points of the Cauchy completion are certain $V$-enriched presheaves $p: C^{op} \to V$, and together form a $V$-category called the Cauchy completion whose homs are the presheaf homs. It is denoted $\bar{C}$.
As we will explain in more detail below, representable presheaves belong to the Cauchy completion, and so the Yoneda embedding of $C$ factors through a full embedding
and we say the $V$-category $C$ is Cauchy-complete if this embedding is an equivalence. We work through a few examples in the following section.
For $C$ a small category write
for the full subcategory of the category of presheaves on $C$ on the retracts of representable functors.
This $\overline{C}$ is called the Cauchy completion of $C$.
For instance (BorceuxDejean, below theorem 1).
The Cauchy completion $\overline{C}$ satisfies the following properties
$\overline{C}$ is a small category;
$C$ is a full subcategory $C \hookrightarrow \overline{C}$;
every idempotent in $\overline{C}$ splits;
the inclusion $C \hookrightarrow \overline{C}$ is an equivalence of categories precisely if already every idempotent in $C$ splits;
there is an equivalence of categories
This appears for instance as (BorceuxDejean, theorem 1).
$\overline{C}$ is small because $[C^{op}, Set]$ is a well-powered category. It contains $C$ as a full subcategory because the Yoneda embedding is a full and faithful functor. Every idempotent splits in $\overline{C}$ because it does so in $[C^{op}, Set]$ and because the composite of two retractions is a retraction.
A retract of a representable $y(c) \in [C^{op}, Set]$ induces an idempotent on $y(c)$ and hence by the Yoneda lemma an idempotent on $c \in C$. If $C$ is already idempotent complete, this splits and produces a retraction of $c$ in $C$ and hence of $y(C)$ in $[C^{op}, Set]$. Since this is necessarily isomorphic to the original retraction, we find that every retract of the representable $y(c)$ is itself representable, therefore $C \simeq \overline{C}$ in this case.
The Cauchy completion $\overline{C}$ is equivalently the full subcategory of $[C^{op}, Set]$ on the tiny objects (“small projective objects”).
This appears for instance as (BorceuxDejean, prop. 2).
The following conditions are equivalent for a small category $C$.
$C$ is Cauchy complete;
$C$ has all small absolute colimits.
We discuss Cauchy completion of small categories in terms of profunctors.
Write $*$ for the terminal category (single object, single morphism). Let $C$ be a small category
The category $C$ is equivalent to the functor category out of the point
Its category of presheaves is equivalent to the profunctor category
In these terms the Yoneda embedding $C \hookrightarrow [C^{op}, Set]$ is the canonical inclusion
Accordingly the Cauchy completion, def. is a full subcategory of the profunctor category
A profunctor $F : * ⇸ C$ belongs to $\overline{C}$ precisely if it has a right adjoint in Prof.
This appears for instance as (BorceuxDejean, prop. 4).
For a small category $C$, the following are equivalent
$C$ is Cauchy complete;
a profunctor $* ⇸ C$ has a right adjoint precisely if it is a functor;
for every small category $A$ a profunctor $A ⇸ C$ has a right adjoint precisely if it is a functor;
This appears for instance as (BorceuxDejean, theorem 2).
In the context of topos theory we say, for $C$ small category, that an adjoint triple of functors
is an essential geometric morphism of toposes $f : Set \to [C,Set]$; or an essential point of $[C,Set]$.
By the adjoint functor theorem this is equivalently simply a single functor $f^* : [C, Set] \to Set$ that preserves all small limits and colimits. Write
for the full subcategory of the functor category on functors that have a left adjoint and a right adjoint.
For $C$ a small category there is an equivalence of categories
of its Cauchy completion, def. , with the category of essential points of $[C,Set]$.
We first exhibit a full inclusion $Topos_{ess}(Set,[C,Set])^{op} \hookrightarrow \overline{C}$.
So let $Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C,Set]$ be an essential geometric morphism. Then because $f_!$ is left adjoint and thus preserves all small colimits and because every set $S \in Set$ is the colimit over itself of the singleton set we have that
is fixed by a choice of copresheaf
The $(f_! \dashv f^*)$-adjunction isomorphism then implies that for all $H \in [C,Set]$ we have
naturally in $H$, and hence that
By assumption this has a further right adjoint $f_\ast$ and hence preserves all colimits. By the discussion at tiny object it follows that $F \in [C,Set]$ is a tiny object. By prop. this means that $F$ belongs to $\overline{C} \subset [C,Set]$.
A morphism $f \Rightarrow g$ between geometric morphisms $f,g : Set \to [C,Set]$ is a geometric transformation, which is a natural transformation $f^* \Rightarrow g^*$, hence by the above a natural transformation $[C,Set](F,-) \Rightarrow [C,Set](G,-)$. By the Yoneda lemma these are in bijection with morphisms $G \to H$ in $[C,Set]$. This gives the full inclusion $Topos_{ess}(Set,[C,Set])^{op} \subset \overline{C}$.
The converse inclusion is now immediate by the same arguments: since the objects in $\overline{C}$ are precisely the tiny objects $F \in [C,Set]$ each of them corresponds to a functor $[C,Set](F,-) : [C,Set] \to Set$ that has a right adjoint. Since this generally also has a left adjoint, it is the inverse image of an essential geometric morphism $f : Set \to [C,Set]$.
Write $Cat_{Cauchy} \hookrightarrow$ Cat for the full sub-2-category of Cat on the Cauchy-complete categories.
The 2-category of Cauchy complete categories is a coreflective full sub-2-category of Topos with essential geometric morphisms
exhibited by the 2-adjunction
We first claim that when working with all categories instead of just the Cauchy-complete categories there is a 2-adjunction
This is exhibited by the following equivalence of hom-categories
natural in $C \in Cat$ and $E \in Topos$. Here
the first equivalence is by definition of essential geometric morphism;
the second equivalence follows by observing that limits and colimits in presheaf categories are computed objectwise;
the third equivalence is again the definition of essential geometric morphisms.
Now by prop. we have that the components of the unit of this adjunction
are equivalences precisely if $C$ is Cauchy-complete. This means that restricted along $Cat_{Cauchy} \hookrightarrow Cat$ the adjunction exhibits a coreflective embedding.
We discuss Cauchy completion in $\mathcal{V}$-enriched category theory, for $\mathcal{V}$ a closed symmetric monoidal category with all limits and colimits. The discussion in ordinary category theory above is the special case where $\mathcal{V} :=$ Set.
The key to the enriched version is the reformulation of ordinary Cauchy completion in terms of profunctors as discussed above. These have an immediate generalization to enriched category theory, and so one takes this formulation as the definition.
As before, we have
Every small $\mathcal{V}$-category $C$ is equivalent to the $\mathcal{V}$-enriched functor category
where $I$ is the $\mathcal{V}$-category with a single object $*$ and $I(*,*) = I_{\mathcal{V}}$, the tensor unit in $\mathcal{V}$.
Also,
and the canonical $\mathcal{V}$-enriched functor
is the enriched Yoneda embedding.
For $C$ a small $\mathcal{V}$-enriched category, the Cauchy completion of $C$ is the full $\mathcal{V}$-subcategory
on those profunctors with a right adjoint in $\mathcal{V}$Prof.
When $\mathcal{V} = \mathbf{Set}$, a $\mathcal{V}$-category is an ordinary category. The Cauchy completion of an ordinary category is its idempotent completion, or Karoubi envelope. This also holds when $\mathcal{V} = \mathbf{Cat}$ or $\mathcal{V} = \mathbf{sSet}$, or more generally whenever $\mathcal{V}$ is a cartesian cosmos where the terminal object is tiny.
When $\mathcal{V} = [0,\infty]$ is the extended nonnegative reals ordered by $\geq$ and with $+$ as monoidal product, $\mathcal{V}$-categories are generalized metric spaces. The Cauchy completion is the usual completion under Cauchy sequences.
When $\mathcal{V} = \mathbf{Ab}$ is abelian groups, a $\mathcal{V}$-category is a pre-additive category. The Cauchy completion is the completion under finite direct sums and idempotent splitting. Notice that there is also a “sub-Cauchy completion” given by completing just under finite direct sums, which turns a pre-additive category into an additive category.
When $\mathcal{V} = \mathbf{Ch}$ is chain complexes, a $\mathcal{V}$-category is a dg-category. Cauchy complete dg-categories are characterized by Nikolić, Street, and Tendas.
When $\mathcal{V} = \mathbf{Slat}$ is the category of sup-lattices, a $\mathcal{V}$-category is a locally posetal, locally cocomplete bicategory, i.e. a quantaloid. The Cauchy completion is some sort of completion under arbitrary sums: it is large even if the original quantaloid is small, and its existence depends on the precise definition we choose of Cauchy completion.
In the $\infty$-categorical context, we can consider enrichment in the $\infty$-category of spectra. The Cauchy completion of an $\infty$-category enriched in spectra is its completion under all finite colimits.
Generalizing to bicategorical enrichment, we can construct from a site $(\mathcal{C}, J)$ a certain bicategory $\mathcal{W}$ such that the Cauchy-complete, symmetric, skeletal $\mathcal{W}$-categories are just the sheaves on $(\mathcal{C}, J)$. Variations on this theme can yield $\mathcal{C}$-indexed categories, stacks, prestacks, or presheaves as Cauchy completions or sub-Cauchy completions for categories enriched in certain bicategories.
Now we look at two examples in more detail: metric spaces and ordinary categories.
We consider first the classical case of metric spaces, but as redefined by Lawvere to mean a category enriched in the poset $V = ([0, \infty], \geq)$, with tensor product given by addition. So, to say $X$ is a Lawvere metric space means that with the set $X$ there is a distance function
such that
for all $x, y, z$ in $X$. (The associativity and identity axioms are here superfluous since $V$ is a poset.) A $V$-enriched functor $f: X \to Y$ here just means a function from $X$ to $Y$ such that
for all $x, y$ in $X$ (again, preservation of composition and of identities is superfluous here), so that $V$-functors are short maps between metric spaces (Lipschitz maps with constant at most $1$). Finally, a $V$-enriched transformation $f \to g: X \to Y$ in this case boils down to an instance of a property: that
for all $x$ in $X$. If $f, g$ are valued in $[0, \infty]$, this just means $f(x) \geq g(x)$ for all $x$.
A point of the Cauchy completion is an $X$-module $p: \mathbf{1} \to X$, i.e., an enriched functor or short map
for which there is an $X$-module $q: X \to \mathbf{1}$ on the other side, an enriched functor
that is right adjoint to $p$ in the sense of modules. This means there is a unit of the adjunction in the bicategory of modules:
and a counit:
Recall now that $Id_X: X \to X$ in the bicategory of modules is the unit bimodule $y_X: X \to V^{X^{op}}$ given by the enriched Yoneda embedding, or in different words, $hom_X = d_X: X^{op} \times X \to [0, \infty]$. Recall also that module composition is defined by a coend formula for a tensor product. If one now tracks through the definitions, keeping in mind that we are in the very simple case of enrichment in a poset, the unit of the adjunction $p \dashv q$ boils down to having the property
and the counit boils down to having the property
To better appreciate what these conditions mean, we point out that $p(x)$ should be thought of as the distance $d_{\bar{X}}(x, p)$ between $x$ and the “ideal point” $p$ in the Cauchy completion $\bar{X}$, and $q(x)$ should be thought of as the companion distance $d_{\bar{X}}(p, x)$. Thus the unit condition above would come down to saying that for every $\varepsilon \gt 0$ there exists $x \in X$ such that
and the counit condition imposes a necessary triangle inequality constraint on the distance functions $p$ and $q$, in order that we get an actual Lawvere metric space $\bar{X}$. If $p, p'$ are two points of the Cauchy completion thus defined, then their distance is defined by the usual formula for enriched presheaves:
It should be noted that even under the classical definition (where we impose symmetry $d(x, y) = d(y, x)$, separation $d(x, y) \gt 0$ for $x \neq y$, and finiteness $d(x,y) \lt \infty$), this provides an elegant alternative definition of Cauchy completion. In essence, all it is doing is taking the metric closure $\bar{X}$ of the embedding of $X$ into the already complete space of short maps:
The presheaf-hom definition of the distance formula for $\bar{X}$, being manifestly non-symmetric, is not the usual definition of distance in the classical symmetric case. However, if we first symmetrize the distance in $[0, \infty]$:
or equivalently
then we do retrieve the classical formula
In other words, the completion $\bar{X}$ of a symmetric metric space $X$ as a general (Lawvere) metric space is not necessarily the same as its completion $\sigma\bar{X}$ as a symmetric metric space, but $\sigma\bar{X}$ is the symmetrisation of $\bar{X}$.
The analysis of Cauchy complete Lawvere metric spaces contains some of the seeds of what happens in other enriched category contexts; the case of ordinary small categories, where the enrichment is no longer in a mere poset but in Set, reflects still more of the phenomena generally associated with Cauchy completions.
Let $C$ be a small category and let the module $p: \mathbf{1} \to C$ be a point of $\bar{C}$, with module $q: C \to \mathbf{1}$ as its right adjoint in the bicategory of modules. As functors,
and the structure of the adjunction is given by unit and counit maps:
As we said in the case of metric spaces, $p(c)$ and $q(c)$ measure “distances” = homs:
The first isomorphism is an instance of the Yoneda lemma, and the second can be seen as follows. The set $q(c)$ is the bimodule composite
where $c$ is shorthand for the module $C(-, c): C^{op} \to Set$; this is just an instance of the Yoneda lemma:
Now using the adjunction $p \dashv q$, there are, for any set $S$, natural bijections
and maps in the bottom line are in bijection with maps $S \to Set^{C^{op}}(p, C(-, c))$. Therefore we have a natural bijection
and this proves $q(c) \cong Set^{C^{op}}(p, C(-, c))$.
With these identifications of $q(c)$ and $p(c)$, the unit of the adjunction $p \dashv q$ takes the form
The coend above is a quotient of
and hence the unit element $\eta$ is represented by a pair of transformations
for some $c$.
Given that, it is now not hard – in fact it is fairly tautological – to verify that on the basis of the triangular equation of the adjunction which says
that
and so a point $p$ in the Cauchy completion $\bar{C}$ must be a retract of a representable $C(-, c)$. Spelling this out a little more: the composite
is an idempotent represented by a morphism $e: c \to c$ in $C$ (by the Yoneda lemma), and this factorization through $p$ splits the idempotent $C(-, e)$ in $Set^{C^{op}}$.
Indeed, the claim is that modules $p: C^{op} \to Set$ in the Cauchy completion are precisely those presheaves on $C$ which arise as retracts of representables in $Set^{C^{op}}$, or in other words may be identified with objects of the idempotent-splitting completion of $C$ (aka the Karoubi envelope of $C$). Therefore, in the $Set$-enriched case, the Cauchy completion is the idempotent-splitting completion. In particular, representables themselves are points of the Cauchy completion.
Notice that in a finitely complete category (such as $Set$ or a presheaf category), idempotents $e: c \to c$ split automatically: just take the equalizer of the pair
For that matter, in any finitely cocomplete category, taking the coequalizer of the above pair would also split the idempotent. Indeed, we can say that idempotents split in a category iff all equalizers of such pairs exist, iff all coequalizers of such pairs exist.
Notice that if $C$ and $D$ are categories, then any functor $F: C \to D$ preserves retracts and therefore splittings of idempotents. Thus, the equalizers above are the sort of limits which are preserved by any functor $F$ whatsoever. They are called absolute limits for that reason. For the same reason, the coequalizers above are absolute colimits: they are precisely the colimits preserved by any functor whatsoever.
Pursuing this a bit further: if $F: C^{op} \to Set$ is any functor, then (because idempotents split in $Set$) there is a unique extension $\bar{F}: \bar{C}^{op} \to Set$ of $F$. Therefore we have an equivalence
and we say that $C$ and $\bar{C}$ are Morita equivalent.
Every ordinary poset clearly is Cauchy complete, since the only idempotents are the identity morphisms. The internalization of this statement requires some extra assumptions:
This appears as (Rosolini, prop. 2.1).
Internal to any exact category the Cauchy completion of any preorder exists and is its poset reflection?.
This appears as (Rosolini, corollary. 2.3).
Moreover, the characterization of Cauchy completion by left adjoint profunctors requires the internal axiom of choice:
In a given ambient context, the following are equivalent:
the axiom of choice holds;
every profunctor $F : A ⇸ C$ between posets is an ordinary functor when it has a right adjoint.
For instance (BorceuxDejean, prop. 5).
David: Concerning the result that on Set the terminal F-coalgebra is the Cauchy completion of the initial F-algebra, for certain F, I wonder if we have to factor completions through the metric space completion, as Barr does in Terminal coalgebras for endofunctors on sets. Perhaps Adamek’s work on Final Algebras are Ideal Completions of Initial Algebras is more natural.
Does this all tie in with the ideal completion as discussed by Awodey where you sum types/sets in a topos into a universal object?
How many kinds of completion are there for an enriched category? I see some may coincide in certain cases.
If two categories can be Morita equivalent, should this be reflected in the page Morita equivalence?
The notion of Cauchy complete categories was introduced in
Bill Lawvere, Metric spaces, generalized logic and closed categories Rend. Sem. Mat. Fis. Milano, 43:135–166 (1973)
Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 (tac)
Surveys are in
Francis Borceux and D. Dejean, Cauchy completion in category theory Cahiers Topologie Géom. Différentielle Catégoriques, 27:133–146, (1986) (numdam)
A. Carboni and Ross Street, Order ideal in categories Pacific J. Math., 124:275–288, 1986.
Further references include for instance
S. R. Johnson, Small Cauchy Completions , JPAA 62 (1989) pp.35-45.
R. Walters, Sheaves and Cauchy complete categories , Cahiers Top. Geom. Diff. Cat. 22 no. 3 (1981) 283-286 (numdam)
R. Walters, Sheaves on sites as Cauchy-complete categories, J. Pure Appl. Algebra 24 (1982) 95-102
Branko Nikolić, Ross Street, Giacomo Tendas, Cauchy completeness for DG-categories, arxiv, 2020
Cauchy completion of internal prosets is discussed in
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