nLab
Cauchy complete category

Contents

Idea

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 CC is small, the Cauchy completion C¯\bar{C} of CC lies between CC and its “free cocompletion”, aka presheaf category

CC¯Set C opC \hookrightarrow \bar{C} \hookrightarrow Set^{C^{op}}

and consists of those presheaves FF dubbed tiny by Lawvere, meaning those presheaves which are connected and projective: the functor

hom Set C op(F,):Set C opSethom_{Set^{C^{op}}}(F, -): Set^{C^{op}} \to Set

preserves small coproducts and coequalizers. All of these concepts generalize straightforwardly to the context of general VV-enriched categories, where VV is a complete, cocomplete, symmetric monoidal closed category.

Lawvere defines a point of the Cauchy completion of a small VV-category CC to be a VV-enriched bimodule p:1Cp: \mathbf{1} \to C (in other words, a VV-functor 1V C op\mathbf{1} \to V^{C^{op}}) for which there is a bimodule q:C1q: C \to \mathbf{1} right adjoint to pp (in the bicategory of enriched bimodules, see profunctor), where 1\mathbf{1} is the unit VV-category. Thus points of the Cauchy completion are certain VV-enriched presheaves p:C opVp: C^{op} \to V, and together form a VV-category called the Cauchy completion whose homs are the presheaf homs. It is denoted C¯\bar{C}.

As we will explain in more detail below, representable presheaves belong to the Cauchy completion, and so the Yoneda embedding of CC factors through a full embedding

i:CC¯i: C \to \bar{C}

and we say the VV-category CC is Cauchy-complete if this embedding is an equivalence. We work through a few examples in the following section.

In ordinary category theory

In terms of splitting of idempotents

Definition

For CC a small category write

C¯[C op,Set] \overline{C} \hookrightarrow [C^{op}, Set]

for the full subcategory of the category of presheaves on CC on the retracts of representable functor.

This C¯\overline{C} is called the Cauchy completion of CC.

For instance (BorceuxDejean, below theorem 1).

Theorem

The Cauchy completion C¯\overline{C} satisfies the following properties

  1. C¯\overline{C} is a small category;

  2. CC is a full subcategory CC¯C \hookrightarrow \overline{C};

  3. every idempotent in C¯\overline{C} splits;

  4. the inclusion CC¯C \hookrightarrow \overline{C} is an equivalence of categories precisely if already every idempotent in CC splits;

  5. there is an equivalence of categories

    [C op,Set][C¯ op,Set]. [C^{op}, Set] \simeq [\overline{C}^{op}, Set] \,.

This appears for instance as (BorceuxDejean, theorem 1).

Proof

C¯\overline{C} is small because [C op,Set][C^{op}, Set] is a well-powered category. It contains CC as a full subcategory because the Yoneda embedding is a full and faithful functor. Every idempotent splits in C¯\overline{C} because it does so in [C op,Set][C^{op}, Set] and because the composite of two retractions is a retraction.

A retract of a representale y(c)[C op,Set]y(c) \in [C^{op}, Set] induces an idempotent on y(c)y(c) and hence by the Yoneda lemma an idempotent on cCc \in C. If CC is already idempotent complete, this splits and produces a retraction of cc in CC and hence of y(C)y(C) in [C op,Set][C^{op}, Set]. Since this is necessarily isomorphic to the original retraction, we find that every retract of the representable y(c)y(c) is itself representable, therefore CC¯C \simeq \overline{C} in this case.

In terms of tiny objects

Proposition

The Cauchy completion C¯\overline{C} is equivalently the full subcategory of [C op,Set][C^{op}, Set] on the tiny objects (“small projective objects”).

This appears for instance as (BorceuxDejean, prop. 2).

In terms of absolute colimits

Proposition

The following conditions are equivalent for a small category CC.

  1. CC is Cauchy complete;

  2. CC has all small absolute colimits.

In terms of profunctors

We discuss Cauchy completion of small categories in terms of profunctors.

Write ** for the terminal category (single object, single morphism). Let CC be a small category

Remark

The category CC is equivalent to the functor category out of the point

CFunc(*,C). C \simeq Func(*,C) \,.

Its category of presheaves is equivalent to the profunctor category

[C op,Set]Profunc(*,C). [C^{op}, Set] \simeq Profunc(*, C) \,.

In these terms the Yoneda embedding C[C op,Set]C \hookrightarrow [C^{op}, Set] is the canonical inclusion

Func(*,C)Profunc(*,C). Func(*,C) \hookrightarrow Profunc(*,C) \,.

Accordingly the Cauchy completion, def. 1 is a full subcategory of the profunctor category

C¯Profunc(*,C). \overline{C} \hookrightarrow Profunc(*,C) \,.
Proposition

A profunctor F:*CF : * ⇸ C belongs to C¯\overline{C} precisely if it has a right adjoint in Prof.

This appears for instance as (BorceuxDejean, prop. 4).

Theorem

For a small category CC, the following are equivalent

  1. CC is Cauchy complete;

  2. a profunctor *C* ⇸ C has a right adjoint precisely if it is a functor;

  3. for every small category AA a profunctor ACA ⇸ C has a right adjoint precisely if it is a functor;

This appears for instance as (BorceuxDejean, theorem 2).

In terms of essential geometric morphisms

In the context of topos theory we say, for CC small category, that an adjoint triple of functors

Setf *f *f ![C,Set] Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C,Set]

is an essential geometric morphism of toposes f:Set[C,Set]f : Set \to [C,Set]; or an essential point of [C,Set][C,Set].

By the adjoint functor theorem this is equivalently simply a single functor f *:[C,Set]Setf^* : [C, Set] \to Set that preserves all small limits and colimits. Write

Topos ess(Set,[C,Set])LRFunc([C,Set],Set)Func([C,Set],Set) Topos_{ess}(Set,[C,Set]) \simeq LRFunc([C,Set], Set) \subset Func([C,Set], Set)

for the full subcategory of the functor category on functors that have a left adjoint and a right adjoint.

Proposition

For CC a small category there is an equivalence of categories

C¯Topos ess(Set,[C,Set]) op \overline{C} \simeq Topos_{ess}(Set, [C,Set])^{op}

of its Cauchy completion, def. 1, with the category of essential points of [C,Set][C,Set].

Proof

We first exhibit a full inclusion Topos ess(Set,[C,Set]) opC¯Topos_{ess}(Set,[C,Set])^{op} \hookrightarrow \overline{C}.

So let Setf *f *f ![C,Set]Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C,Set] be an essential geometric morphism. Then because f !f_! is left adjoint and thus preserves all small colimits and because every set SSetS \in Set is the colimit over itself of the singleton set we have that

f !S sSf !(*) f_! S \simeq \coprod_{s \in S} f_!(*)

is fixed by a choice of copresheaf

F:=f !(*)[C,Set]. F := f_!(*) \in [C, Set] \,.

The (f !f *)(f_! \dashv f^*)-adjunction isomorphism then implies that for all H[C,Set]H \in [C,Set] we have

f *HSet(*,f *H)[C,Set](f !*,H)[C,Set](F,H). f^* H \simeq Set(*, f^* H) \simeq [C,Set](f_! *, H) \simeq [C,Set](F,H) \,.

naturally in HH, and hence that

f *()[C,Set](F,):Set[C,Set]. f^*(-) \simeq [C,Set](F,-) : Set \to [C,Set] \,.

By assumption this has a further right adjoint f *f_\ast and hence preserves all colimits. By the discussion at tiny object it follows that F[C,Set]F \in [C,Set] is a tiny object. By prop. 1 this means that FF belongs to C¯[C,Set]\overline{C} \subset [C,Set].

A morphism fgf \Rightarrow g between geometric morphisms f,g:Set[C,Set]f,g : Set \to [C,Set] is a geometric transformation, which is a natural transformation f *g *f^* \Rightarrow g^*, hence by the above a natural transformation [C,Set](F,)[C,Set](G,)[C,Set](F,-) \Rightarrow [C,Set](G,-). By the Yoneda lemma these are in bijection with morphisms GHG \to H in [C,Set][C,Set]. This gives the full inclusion Topos ess(Set,[C,Set]) opC¯Topos_{ess}(Set,[C,Set])^{op} \subset \overline{C}.

The converse inclusion is now immediate by the same arguments: since the objects in C¯\overline{C} are precisely the tiny objects F[C,Set]F \in [C,Set] each of them corresponds to a functor [C,Set](F,):[C,Set]Set[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[C,Set]f : Set \to [C,Set].

Write Cat CauchyCat_{Cauchy} \hookrightarrow Cat for the full sub-2-category of Cat on the Cauchy-complete categories.

Proposition

The 2-category of Cauchy complete categories is a coreflective full sub-2-category of Topos with essential geometric morphisms

Cat CauchyTopos ess Cat_{Cauchy} \hookrightarrow Topos_{ess}

exhibited by the 2-adjunction

([,Set]Topos ess(Set,)):Topos essTopos ess(Set,)Cat Cauchy(,Set)Cat Cauchy. ([-,Set] \dashv Topos_{ess}(Set,-) ) : Topos_{ess} \stackrel{\overset{Cat_{Cauchy}(-,Set)}{\hookleftarrow}}{\underset{Topos_{ess}(Set,-)}{\to}} Cat_{Cauchy} \,.
Proof

We first claim that when working with all categories instead of just the Cauchy-complete categories there is a 2-adjunction

([,Set]Topos ess(Set,)):Topos essTopos ess(Set,)[,Set]Cat. ([-,Set] \dashv Topos_{ess}(Set,-) ) : Topos_{ess} \stackrel{\overset{[-,Set]}{\leftarrow}}{\underset{Topos_{ess}(Set,-)}{\to}} Cat \,.

This is exhibited by the following equivalence of hom-categories

Func(C,Topos ess(Set,E)) Func(C,LRFunc(E,Set)) LRFunc(E,Func(C,Set))=:LRFunc(E,[C,Set]) Topos ess([C,Set],E) \begin{aligned} Func(C, Topos_{ess}(Set, E)) & \simeq Func(C, LRFunc(E, Set)) \\ & \simeq LRFunc(E, Func(C,Set)) =: LRFunc(E, [C,Set]) \\ & \simeq Topos_{ess}([C,Set], E) \end{aligned}

natural in CCatC \in Cat and EToposE \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. 4 we have that the components of the unit of this adjunction

CTopos ess(Set,[C,Set]) C \to Topos_{ess}(Set,[C,Set])

are equivalences precisely if CC is Cauchy-complete. This means that restricted along Cat CauchyCatCat_{Cauchy} \hookrightarrow Cat the adjunction exhibits a coreflective embedding.

In enriched category theory

We discuss Cauchy completion in 𝒱\mathcal{V}-enriched category theory, for 𝒱\mathcal{V} a closed symmetric monoidal category with all limits and colimits. The discussion in ordinary category theory above is the special case where 𝒱:=\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

Remark

Every small 𝒱\mathcal{V}-category CC is equivalence to the 𝒱\mathcal{V}-enriched functor category

C𝒱Func(I,C), C \simeq \mathcal{V}Func(I,C) \,,

where II is the 𝒱\mathcal{V}-category with a single object ** and I(*,*)=I 𝒱I(*,*) = I_{\mathcal{V}}, the tensor unit in 𝒱\mathcal{V}.

Also,

[C op,𝒱]𝒱Profunc(I,C) [C^{op}, \mathcal{V}] \simeq \mathcal{V}Profunc(I, C)

and the canonical 𝒱\mathcal{V}-enriched functor

𝒱Func(I,C)𝒱Profuc(I,C) \mathcal{V}Func(I,C) \hookrightarrow \mathcal{V}Profuc(I,C)

is the enriched Yoneda embedding.

Definition

For CC a small 𝒱\mathcal{V}-enriched category, the Cauchy completion of CC is the full 𝒱\mathcal{V}-subcategory

C¯𝒱Profunc(I,C) \overline{C} \hookrightarrow \mathcal{V}Profunc(I,C)

on those profunctors with a right adjoint in 𝒱\mathcal{V}Prof.

Examples

  • When 𝒱=Set\mathcal{V} = \mathbf{Set}, a 𝒱\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 𝒱=Cat\mathcal{V} = \mathbf{Cat} or 𝒱=sSet\mathcal{V} = \mathbf{sSet}, or more generally whenever 𝒱\mathcal{V} is a cartesian cosmos where the terminal object is tiny.

  • When 𝒱=[0,]\mathcal{V} = [0,\infty] is the extended nonnegative reals ordered by \geq and with ++ as monoidal product, 𝒱\mathcal{V}-categories are generalized metric spaces. The Cauchy completion is the usual completion under Cauchy sequences.

  • When 𝒱=Ab\mathcal{V} = \mathbf{Ab} is abelian groups, a 𝒱\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 𝒱=Slat\mathcal{V} = \mathbf{Slat} is the category of sup-lattices, a 𝒱\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 (𝒞,J)(\mathcal{C}, J) a certain bicategory 𝒲\mathcal{W} such that the Cauchy-complete, symmetric, skeletal 𝒲\mathcal{W}-categories are just the sheaves on (𝒞,J)(\mathcal{C}, J). Variations on this theme can yield 𝒞\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.

Metric spaces

We consider first the classical case of metric spaces, but as redefined by Lawvere to mean a category enriched in the poset V=([0,],)V = ([0, \infty], \geq), with tensor product given by addition. So, to say XX is a Lawvere metric space means that with the set XX there is a distance function

d X=hom X:X×X[0,]d_X = hom_X: X \times X \to [0, \infty]

such that

d(x,y)+d(y,z)d(x,z),0d(x,x)d(x, y) + d(y, z) \geq d(x, z), \qquad 0 \geq d(x, x)

for all x,y,zx, y, z in XX. (The associativity and identity axioms are here superfluous since VV is a poset.) A VV-enriched functor f:XYf: X \to Y here just means a function from XX to YY such that

d X(x,y)d Y(f(x),f(y))d_X(x, y) \geq d_Y(f(x), f(y))

for all x,yx, y in XX (again, preservation of composition and of identities is superfluous here), so that VV-functors are short maps between metric spaces (Lipschitz maps with constant at most 11). Finally, a VV-enriched transformation fg:XYf \to g: X \to Y in this case boils down to an instance of a property: that

0d Y(f(x),g(x))0 \geq d_Y(f(x), g(x))

for all xx in XX. If f,gf, g are valued in [0,][0, \infty], this just means f(x)g(x)f(x) \geq g(x) for all xx.

A point of the Cauchy completion is an XX-module p:1Xp: \mathbf{1} \to X, i.e., an enriched functor or short map

p:X op[0,]p: X^{op} \to [0, \infty]

for which there is an XX-module q:X1q: X \to \mathbf{1} on the other side, an enriched functor

q:X[0,]q: X \to [0, \infty]

that is right adjoint to pp in the sense of modules. This means there is a unit of the adjunction in the bicategory of modules:

(Id:11)(1pXq1)(Id: \mathbf{1} \to \mathbf{1}) \to (\mathbf{1} \overset{p}{\to} X \overset{q}{\to} \mathbf{1})

and a counit:

(Xq1pX)(Id:XX)(X \overset{q}{\to} \mathbf{1} \overset{p}{\to} X) \to (Id: X \to X)

Recall now that Id X:XXId_X: X \to X in the bicategory of modules is the unit bimodule y X:XV X opy_X: X \to V^{X^{op}} given by the enriched Yoneda embedding, or in different words, hom X=d X:X op×X[0,]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 pqp \dashv q boils down to having the property

0 xXq(x)+p(x)=inf xXq(x)+p(x)0 \geq \int^{x \in X} q(x) + p(x) = \inf_{x \in X} q(x) + p(x)

and the counit boils down to having the property

p(x)+q(y)d X(x,y)p(x) + q(y) \geq d_X(x, y)

To better appreciate what these conditions mean, we point out that p(x)p(x) should be thought of as the distance d X¯(x,p)d_{\bar{X}}(x, p) between xx and the “ideal point” pp in the Cauchy completion X¯\bar{X}, and q(x)q(x) should be thought of as the companion distance d X¯(p,x)d_{\bar{X}}(p, x). Thus the unit condition above would come down to saying that for every ε>0\varepsilon \gt 0 there exists xXx \in X such that

d X¯(p,x)<ε,d X¯(x,p)<εd_{\bar{X}}(p, x) \lt \varepsilon, \, d_{\bar{X}}(x, p) \lt \varepsilon

and the counit condition imposes a necessary triangle inequality constraint on the distance functions pp and qq, in order that we get an actual Lawvere metric space X¯\bar{X}. If p,pp, p' are two points of the Cauchy completion thus defined, then their distance is defined by the usual formula for enriched presheaves:

d(p,p)= xXhom [0,](p(x),p(x))=sup xXmax{0,p(x)p(x)}d(p, p') = \int_{x \in X} \hom_{[0, \infty]}(p(x), p'(x)) = \sup_{x \in X} \max\{0, p'(x) - p(x)\}

It should be noted that even under the classical definition (where we impose symmetry d(x,y)=d(y,x)d(x, y) = d(y, x), separation d(x,y)>0d(x, y) \gt 0 for xyx \neq y, and finiteness d(x,y)<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 X¯\bar{X} of the embedding of XX into the already complete space of short maps:

y X:X[0,] X op:xd X(,x)y_X: X \to [0, \infty]^{X^{op}}: x \mapsto d_X(-, x)

The presheaf-hom definition of the distance formula for X¯\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,][0, \infty]:

σd(r,s)=d(r,s)+d(s,r)=max{0,sr}+max{0,rs}=|rs|\sigma d(r, s) = d(r, s) + d(s, r) = \max\{0, s-r\} + \max\{0, r-s\} = |r - s|

or equivalently

σd(r,s)=d(r,s)+d(s,r)=max(max{0,sr},max{0,rs})=|rs|\sigma d(r, s) = d(r, s) + d(s, r) = \max(\max\{0, s-r\}, \max\{0, r-s\}) = |r - s|

then we do retrieve the classical formula

d(p,p)= xXσd(p(x),p(x))=sup xX|p(x)p(x)|d(p, p') = \int_{x \in X} \sigma d(p(x), p'(x)) = \sup_{x \in X} |p(x) - p'(x)|

In other words, the completion X¯\bar{X} of a symmetric metric space XX as a general (Lawvere) metric space is not necessarily the same as its completion σX¯\sigma\bar{X} as a symmetric metric space, but σX¯\sigma\bar{X} is the symmetrisation of X¯\bar{X}.

Ordinary (SetSet-enriched) categories

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 CC be a small category and let the module p:1Cp: \mathbf{1} \to C be a point of C¯\bar{C}, with module q:C1q: C \to \mathbf{1} as its right adjoint in the bicategory of modules. As functors,

p:C opSet,q:CSetp: C^{op} \to Set, \, q: C \to Set

and the structure of the adjunction is given by unit and counit maps:

η:1 cOb(C)q(c)×p(c),ε c,d:p(c)×q(d)C(c,d)\eta: 1 \to \int^{c \in Ob(C)} q(c) \times p(c), \qquad \varepsilon_{c, d}: p(c) \times q(d) \to C(c, d)

As we said in the case of metric spaces, p(c)p(c) and q(c)q(c) measure “distances” = homs:

p(c)Set C op(C(,c),p),q(c)Set C op(p,C(,c))p(c) \cong Set^{C^{op}}(C(-, c), p), \qquad q(c) \cong Set^{C^{op}}(p, C(-, c))

The first isomorphism is an instance of the Yoneda lemma, and the second can be seen as follows. The set q(c)q(c) is the bimodule composite

(1cCq1)(\mathbf{1} \overset{c}{\to} C \overset{q}{\to} \mathbf{1})

where cc is shorthand for the module C(,c):C opSetC(-, c): C^{op} \to Set; this is just an instance of the Yoneda lemma:

q CC(,c)=def dCq(d)×C(d,c)Yonedaq(c).q \circ_C C(-, c) \overset{def}{=} \int^{d \in C} q(d) \times C(d, c) \overset{Yoneda}{\cong} q(c).

Now using the adjunction pqp \dashv q, there are, for any set SS, natural bijections

Sq(c)Sq CC(,c)p 1SC(,c)p()×SC(,c)\frac{\frac{S \to q(c)}{S \to q \circ_C C(-, c)}}{\frac{p \circ_{\mathbf{1}} S \to C(-, c)}{p(-) \times S \to C(-, c)}}

and maps in the bottom line are in bijection with maps SSet C op(p,C(,c))S \to Set^{C^{op}}(p, C(-, c)). Therefore we have a natural bijection

Sq(c)SSet C op\frac{S \to q(c)}{S \to Set^{C^{op}}}

and this proves q(c)Set C op(p,C(,c))q(c) \cong Set^{C^{op}}(p, C(-, c)).

With these identifications of q(c)q(c) and p(c)p(c), the unit of the adjunction pqp \dashv q takes the form

η:1 cSet C op(p,C(,c))×Set C op(C(,c),p)\eta: 1 \to \int^{c} Set^{C^{op}}(p, C(-, c)) \times Set^{C^{op}}(C(-, c), p)

The coend above is a quotient of

cSet C op(p,C(,c))×Set C op(C(,c),p)\sum_c Set^{C^{op}}(p, C(-, c)) \times Set^{C^{op}}(C(-, c), p)

and hence the unit element η\eta is represented by a pair of transformations

i:pC(,c),π:C(,c)pi: p \to C(-, c), \qquad \pi: C(-, c) \to p

for some cc.

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

(ppηpqpεp)=1 p,(p \overset{p \circ \eta}{\to} p \circ q \circ p \overset{\varepsilon}{\to} p) = 1_p,

that

(piC(,c)πp)=1 p(p \overset{i}{\to} C(-, c) \overset{\pi}{\to} p) = 1_p

and so a point pp in the Cauchy completion C¯\bar{C} must be a retract of a representable C(,c)C(-, c). Spelling this out a little more: the composite

C(,c)πpiC(,c)C(-, c) \overset{\pi}{\to} p \overset{i}{\to} C(-, c)

is an idempotent represented by a morphism e:cce: c \to c in CC (by the Yoneda lemma), and this factorization through pp splits the idempotent C(,e)C(-, e) in Set C opSet^{C^{op}}.

Indeed, the claim is that modules p:C opSetp: C^{op} \to Set in the Cauchy completion are precisely those presheaves on CC which arise as retracts of representables in Set C opSet^{C^{op}}, or in other words may be identified with objects of the idempotent-splitting completion of CC (aka the Karoubi envelope of CC). Therefore, in the SetSet-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 SetSet or a presheaf category), idempotents e:cce: c \to c split automatically: just take the equalizer of the pair

c1ecc \stackrel{\overset{e}{\to}}{\underset{1}{\to}} c

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 CC and DD are categories, then any functor F:CDF: 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 FF 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 opSetF: C^{op} \to Set is any functor, then (because idempotents split in SetSet) there is a unique extension F¯:C¯ opSet\bar{F}: \bar{C}^{op} \to Set of FF. Therefore we have an equivalence

Set C opSet C¯ opSet^{C^{op}} \simeq Set^{\bar{C}^{op}}

and we say that CC and C¯\bar{C} are Morita equivalent.

Of prosets

Every ordinary poset clearly is Cauchy complete, since the only idempotents are the identity morphisms. The internalization of this statement requires some extra assumptions:

Proposition

Internal to any regular category every poset is Cauchy complete.

This appears as (Rosolini, prop. 2.1).

Proposition

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:

Proposition

In a given ambient context, the following are equivalent:

  1. the axiom of choice hold;

  2. every profunctor F:ACF : A &#x21F8; C between posets is an ordinary functor when it has a right adjoint.

For instance (BorceuxDejean, prop. 5).

Discussion

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?

References

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

  • 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

Cauchy completion of internal prosets is discussed in

  • G. Rosolini, A note on Cauchy completeness for preorders (pdf)

Revised on August 10, 2014 06:36:01 by Urs Schreiber (185.37.147.14)