nLab
compact object

Idea

An object of a category is called compact if it is “finite” or “small” in some precise sense. There are however different formalizations of this idea. Here discussed is the notion, usually going by this term, where an object $X$ is called compact if mapping out of it commutes with filtered colimits.

This means that if any other object $A$ is given as the colimit of a “suitably increasing” family of objects $\{A_i\}$, then every morphism

$$X\to A=\underset{{\to }_i}{\lim }{A}_i$$X \to A = \lim_{\to_i} A_i

out of the compact object $X$ into that colimit factors through one of the inclusions $A_i\to \underset{{\to }_i}{\lim }{A}_i$.

The notion of small object is essentially the same, with a bit more flexibility on when the family $\{A_i\}$ is taken to be “suitably increasing”. An important application of the above factorization property is accordingly named the small object argument. On the other hand, there is also the notion of finite object (in a topos) which, while closely related, is different. See also Subtleties and different meanings below.

Definition

Definition

Let $C$ be a locally small category that admits filtered colimits. Then an object $X\in C$ is compact, (or finitely presented or finitely presentable or of finite presentation), if the corepresentable functor

$${\mathrm{Hom}}_C(X,-):C\to \mathrm{Set}$$Hom_C(X,-) : C \to Set

preserves these filtered colimits. This means that for every filtered category $D$ and every functor $F:D\to C$, the canonical morphism

$$\underset{{\to }_d}{\lim }C(X,F(d))\stackrel{\simeq }{\to }C(X,\underset{{\to }_d}{\lim }F(d))$$\underset{\to_d}{\lim} C(X,F(d)) \stackrel{\simeq}{\to} C(X, \underset{\to_d}{\lim} F(d))

is an isomorphism.

More generally, if $\kappa$ is a regular cardinal, then an object $X$ such that $C(X,-)$ commutes with $\kappa$-filtered colimits is called $\kappa$-compact, or $\kappa$-presented or $\kappa$-presentable. An object which is $\kappa$-compact for some regular $\kappa$ is called a small object.

Properties

Proposition

A $\kappa$-small colimit of $\kappa$-compact objects is again a $\kappa$-compact object.

Proof

Let $D$ be a $\kappa$-small category and $X:D\to C$ a diagram of $\kappa$-compact objects. Let $I$ be a $\kappa$-filtered category and $A:I\to C$ a $\kappa$-filtered diagram in $C$. Then

$$\mathrm{Hom}(\underset{{\to }_d}{\lim }{X}_d,\underset{{\to }_i}{\lim }{A}_i)\simeq \underset{{\leftarrow }_d}{\lim }\mathrm{Hom}(X_d,\underset{{\to }_i}{\lim }{A}_i)$$Hom(\lim_{\to_d} X_d, \lim_{\to_i} A_i) \simeq \lim_{\leftarrow_d} Hom(X_d, \lim_{\to_i} A_i)

by general properties of the hom functor. Now using that every $X_d$ is $\kappa$-compact and $I$ is $\kappa$-filtered this is

$$\cdots \simeq \underset{{\leftarrow }_d}{\lim }\underset{{\to }_i}{\lim }\mathrm{Hom}(X_d,A_i).$$\cdots \simeq \lim_{\leftarrow_d} \lim_{\to_i} Hom(X_d, A_i) \,.

Since this (co)limit is taken in Set ,the $\kappa$-small limit over $D$ commutes with the $\kappa$-filtered colimit

$$\cdots \simeq \underset{{\to }_i}{\lim }\underset{{\leftarrow }_d}{\lim }\mathrm{Hom}(X_d,A_i).$$\cdots \simeq \lim_{\to_i} \lim_{\leftarrow_d} Hom(X_d, A_i) \,.

We can take the limit again to a colimit in the first argument

$$\cdots \simeq \underset{{\to }_i}{\lim }\mathrm{Hom}(\underset{{\to }_d}{\lim }{X}_d,A_i),$$\cdots \simeq \lim_{\to_i} Hom(\lim_{\to_d} X_d, A_i) \,,

which proves the claim.

Examples

• In $C=$ Set an object is compact precisely if it is a finite set. For this to hold constructively, filtered categories (appearing in the definition of filtered colimit) have to be understood as categories admitting cocones of every Bishop-finite diagram. (An object of Set is a Kuratowski-finite precisely if it is a finitely generated object, or equivalently if it is compact when regarded as a discrete topological space.)

• For $C$ a topos, $X$ is compact if …

• In $C=$ Grp an object is compact precisely if it is finitely presented? as a group.

• More generally, if $C$ is any variety of algebras, then an object is compact precisely if it is finitely presented? as an algebra. A proof can be found in Corollary 3.13 of LPAC.

• Let $X$ be a topological space and let $C=\mathrm{Op}(X)$ be the category of open subsets of $X$. Then an open subset $U\in C$ is a compact object in $C$ precisely if it is a compact topological space. (It is not true that $X$ is a compact object of $\mathrm{Top}$ iff it is a compact topological space; see below.)

Subtleties and different meanings

One has to be careful about the following variations of the theme of compactness.

(Some of these subtleties are resolved by noticing that there is a hierarchy of notions of compact objects that are secretly different but partly go by the same name. Some discussion of this is currently at compact topos, but more detailed discussion should eventually be somewhere…)

Compactness in additive categories

When $C$ is an additive category (often a triangulated category), an object $x$ in $C$ is called compact if for every set $S$ of objects of $C$ such that the coproduct ${\coprod }_{s\in S}s$ exists, the canonical map

$$\coprod _{s\in S}C(x,s)\to C(x,\coprod _{s\in S}s)$$\coprod_{s\in S} C(x,s)\to C(x,\coprod_{s\in S}s)

is an isomorphism of commutative monoids.

Here is an application of this concept to characterize which abelian categories are categories of modules of some ring:

Theorem

Let $C$ be an abelian category. If $C$ has all small coproducts and has a compact projective generator, then $C\simeq R\mathrm{Mod}$ for some ring $R$. In fact, in this situation we can take $R=C(x,x{)}^{\mathrm{op}}$ where $x$ is any compact projective generator. Conversely, if $C\simeq R\mathrm{Mod}$, then $C$ has all small coproducts and $x=R$ is a compact projective generator.

Proof

This theorem, minus the explicit description of $R$, can be found as Exercise F on page 103 of Peter Freyd’s book Abelian Categories. The first part of this theorem can also be found as Prop. 2.1.7. of Victor Ginzburg’s Lectures on noncommutative geometry. Conversely, it is easy to see that $R$ is a compact projective generator of $R\mathrm{Mod}$.

Zoran: While Ginzburg’s reference is surely a worthy to look at, it would be better not to give false impression that this reconstruction theorem is due Ginzburg or at all new. It is rather a classical and well know fact probably from early 1960s, essentially small strengthening of a variant of a circle of abelian reconstruction theorems including the Gabriel-Popescu theorem(probably our variant could be read off from classical algera book by Faith for example, or Popescu’s book on abelian categories, in any case it is well known in noncommutative algebraic geometry). In fact for this fact, if I think better, the reconstruction belongs usually to expositions which treat classical Morita theory for rings.

A triangulated category is compactly generated if it is generated (see generator) by a set of compact objects.

The notion can be modified for categories enriched over a closed monoidal category (compare to the notions of finite and/or rigid objects in various contexts).

Compact objects in the derived categories of quasicoherent sheaves over a scheme are called perfect complexes. Any compact object in the category of modules over a perfect ring is finitely generated as a module. Lurie uses $\kappa$-compact objects in the setup of $(\mathrm{\infty },1)$-categories.

In non-additive contexts, the above definition is not right. For instance, with this definition a topological space would be compact iff it is connected. In general one should expect to instead preserve filtered colimits, as above.

Compact objects in $\mathrm{Top}$

Recall the above example of compact topological spaces. Notice that the statement which one might expect, that a topological space $X$ is compact if it is a compact object in Top, is not quite right in general.

A counterexample is given for instance on page 49 of Hovey’s Model Categories, which itself was corrected by Don Stanley (see the errata of that book). See also the blog discussion here.

Namely, the two-element set with the indiscrete topology is a compact space $X$ for which

(1)$$\mathrm{Hom}(X,-):\mathrm{Top}\to \mathrm{Top}$$Hom(X, -): Top \rightarrow Top

doesn’t preserve filtered colimits, in fact not even colimits of sequences (functors out of the ordered set of natural numbers).

For example, consider the sequence of spaces

(2)$$X_n=[n,\mathrm{\infty })\times \{0,1\}$$X_n=[n,\infty) \times \{0,1\}

where the open sets are of the form

(3)$$[n,\mathrm{\infty }]\cup [m,\mathrm{\infty })\times \{1\}$$[n, \infty]\cup [m,\infty) \times \{1\}

(where $m\ge n$), plus the empty set. Define $X_n\to X_{n+1}$ so that it sends a pair $(k,ϵ)$ to itself if $k>n$, and $(n,ϵ)$ to $(n+1,ϵ)$. This defines a functor

(4)$$F:\mathbb{N}\to \mathrm{Top}$$F: \mathbb{N} \rightarrow Top

The colimit $X_{\mathrm{\infty }}$ of this sequence is the two-element set $\{0,1\}$ with the indiscrete topology. However, the identity map on this space does not factor through any of the canonical maps $X_n\to X_{\mathrm{\infty }}$. It follows that the comparison map

(5)$${\mathrm{colim}}_n\mathrm{Hom}(X_{\mathrm{\infty }},X_n)\to \mathrm{Hom}(X_{\mathrm{\infty }},X_{\mathrm{\infty }})$$colim_n Hom(X_\infty, X_n) \rightarrow Hom(X_\infty, X_\infty)

is not surjective, and therefore not an isomorphism.

Todd (posted from n-category cafe): I don’t know if the story is any different for $X$ compact Hausdorff, but it could be worth considering.

But with a bit of care on the assumptions, similar results do hold:

If $Y$ is compact, then $\mathrm{hom}(Y,-)$ preserves colimits of functors mapping out of limit ordinals, provided that the arrows of the cocone diagram,

(6)$$X_{\alpha }\to X_{\beta },$$X_\alpha \rightarrow X_\beta,

are closed inclusions of $T_1$ spaces. (This applies for example to the sequence of inclusions of n-skeleta in a CW-complex. Taking $Y=S_k$, this has obvious desirable consequences for the functor ${\pi }_k$.)

This example is discussed on page 50 of Hovey’s book.

Hovey wants this result in view of a small object argument on the way to proving that $\mathrm{Top}$ is a model category.

Related concepts

• compact element (in a (0,1)-category)

• compact object

• compact object in an (∞,1)-category

• small object, small object argument

• finitely generated object

• locally presentable category

References

Compact objects are discussed under the term “finitely presentable” or “finitely-presentable” objects for instance in

• Locally Presentable and Accessible Categories

• Kashiwara, Schapira, Categories and Sheaves, Definition 6.3.2;

• Peter Johnstone, Stone Spaces, Definition VI.1.8.

For the pages quoted in the context of the discussion of compact objects in Top see

• Mark Hovey, Model categories.

For the general definition with an eye towards the definition of compact object in an (infinity,1)-category see section A.1.1 section 5.3.4 of

• Jacob Lurie, Higher Topos Theory