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 preserves 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
out of the compact object $X$ into that colimit factors through one of the coprojections $A_i \to \underset{\underset{i}{\longrightarrow}}{lim A_i}$.
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.
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
preserves these filtered colimits. This means that for every filtered category $D$ and every functor $F : D \to C$, the canonical morphism
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.
(smooth colimits of compact objects are compact)
A $\kappa$-small colimit of $\kappa$-compact objects is again a $\kappa$-compact object.
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
by general properties of the hom functor. Now using that every $X_d$ is $\kappa$-compact and $I$ is $\kappa$-filtered this is
Since this (co)limit is taken in Set, the $\kappa$-small limit over $D$ commutes with the $\kappa$-filtered colimit
Finally we make take the limit again to a colimit in the first argument
which yields the claim.
In a $\lambda$-accessible category, if $\lambda$ is sharply smaller than $\kappa$, then every $\kappa$-compact object is a retract of a $\kappa$-small $\lambda$-filtered colimit of $\lambda$-compact objects.
See Adamek-Rosicky, Remark 2.15. It is noted there that with the more technical proof of Makkai-Pare, Proposition 2.3.11 the words “a retract of” can be omitted.
If we weaken the hypothesis to $\lambda\le \kappa$, then we retain all of the result except for $\lambda$-filteredness of the colimit.
In a locally $\lambda$-presentable category, if $\lambda\le \kappa$, then every $\kappa$-compact object is a retract of a $\kappa$-small colimit of $\lambda$-compact objects.
See Adamek-Rosicky, Remark 1.30. It is claimed there that the words “a retract of” can be omitted by reference to an argument in Makkai-Pare, but it seems unclear how this argument is intended to be used. An alternative proof of this improvement is proposed at this mathoverflow question.
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:
$C$ is a sheaf topos on a site whose topology is generated by finite covering families and $X$ is a representable sheaf;
in particular if $C$ is a coherent topos and $X$ is a coherent object.
However, there exist compact objects which are not coherent, c.f. the Elephant, D3.3.12.
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 may be found in Adámek-Rosický 94, Corollary 3.13.
Let $X$ be a topological space and let $C = Op(X)$ 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 $Top$ iff it is a compact topological space; see below.)
A finite-dimensional vector space is compact in Vect, see here.
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…)
In the Elephant, what Johnstone calls compact objects are those objects such that the top element of the poset of subobjects $\operatorname{Sub}(C)$ is a compact element; he reserves the term finitely-presented for the notion of compact on this page.
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
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:
Let $C$ be an abelian category. If $C$ has all small coproducts and has a compact projective generator, then $C \simeq R Mod$ for some ring $R$. In fact, in this situation we can take $R = C(x,x)^{op}$ where $x$ is any compact projective generator. Conversely, if $C \simeq R Mod$, then $C$ has all small coproducts and $x = R$ is a compact projective generator.
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 $R Mod$.
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.
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.
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 in Hovey (1999), page 49, 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
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
where the open sets are of the form
(where $m \geq n$), plus the empty set. Define $X_n \rightarrow X_{n+1}$ so that it sends a pair $(k, \epsilon)$ to itself if $k \gt n$, and $(n,\epsilon)$ to $(n+1,\epsilon)$. This defines a functor
The colimit $X_\infty$ 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 $X_n \rightarrow X_\infty$. It follows that the comparison map
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 $Y \in Top$ is a compact topological space, then $hom(Y,-)$ preserves colimits of functors mapping out of limit ordinals, provided that the arrows of the cocone diagram,
are closed inclusions of $T_1$-spaces, and $X_\beta$ is the colimit of earlier $X_\alpha$ if $\beta$ is itself a limit ordinal.
This applies for example to the sequence of inclusions of $n$-skeleta in a CW-complex. By considering the cases $Y=S^k$ and $Y = S^k \times I$, this implies that the functor $\pi_k$ also preserves colimits of such sequences. Applications also occur in the theory of classifying bundles. For example, the total space $EO(n)$ of the $O(n)$-universal principal bundle can be constructed as the colimit $\underset{k \to \infty}{colim} \; V_n(\mathbb{R}^{n+k})$ of Stiefel manifolds of orthonormal $n$-frames in $\mathbb{R}^{n+k}$. To show $EO(n)$ is weakly contractible, it suffices, by this result, to show that for each $j$, the homotopy group $\pi_j(V_n(\mathbb{R}^{n+k}))$ vanishes for sufficiently large $k$.
Let $X$ denote the colimit of the diagram mapping out of the limit ordinal $\kappa$. We must show that every map $f \colon Y \to X$ factors through one of the universal cocone components $i_\alpha \colon X_\alpha \to X$, where $\alpha \lt \kappa$. Suppose not. Starting from any $\alpha \lt \kappa$, pick a point $x_0 \in f(Y)$ not in the image $i_\alpha(X_\alpha)$. There is a minimal $\alpha_1 \lt \kappa$ such that $x_0 \in i_{\alpha_1}(X_{\alpha_1})$. Proceeding inductively, choose $x_n \in f(Y)$ such that $x_n \notin i_{\alpha_n}(X_{\alpha_n})$, and let $\alpha_{n+1}$ be the minimal ordinal such that $x_n \in i_{\alpha_{n+1}}(X_{\alpha_{n+1}})$. Then let $\beta$ be the limit of the $\alpha_n$, and let $S$ be the set $\{x_n: n \in \mathbb{N}\}$, so that $S \subseteq i_\beta(X_\beta)$. Any subset $K$ of $S$ is closed, because closure of $K \subseteq i_\beta(X_\beta)$ is equivalent to closure of of each $K \cap i_{\alpha_n}(X_{\alpha_n})$ (since that is how the colimit topology on $X_\beta$ is defined), and all these intersections are finite and therefore closed by the $T_1$ condition. It follows that $f^{-1}(S)$ is an infinite discrete subspace of $Y$, but this contradicts compactness.
This proof is essentially that given in Hovey (1999), page 50, where this result is needed towards the small object argument on the way to establishing the classical model structure on topological spaces, see this lemma.
compact element (in a (0,1)-category)
geometry | monoidal category theory | category theory |
---|---|---|
perfect module | (fully-)dualizable object | compact object |
Compact objects are discussed under the term “finitely presentable” or “finitely-presentable” objects for instance in
Jiří Adámek, Jiří Rosický, Locally Presentable and Accessible Categories, Cambridge University Press in the London Mathematical Society Lecture Note Series, number 189, (1994)
Michael Makkai, Robert Paré, Accessible categories: The foundations of categorical model theory Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.
Masaki Kashiwara, Pierre Schapira, around Definition 6.3.3 of Categories and Sheaves;
Peter Johnstone, Stone Spaces, Definition VI.1.8;
Peter Johnstone, the Elephant, D2.3.1.
For the pages quoted in the context of the discussion of compact objects in Top see
.
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
Last revised on June 29, 2024 at 09:23:54. See the history of this page for a list of all contributions to it.