nLab
skeleton

Definition

A category is skeletal if objects that are isomorphic are necessarily equal. (This is an irredeemably evil notion.) A skeleton of a category is any skeletal subcategory whose inclusion functor exhibits it as equivalent to the original category.

If the axiom of choice holds, then every category has a skeleton: simply choose one object in each isomorphism class. In fact, the statement that every (possibly small) category has a skeleton is equivalent to the axiom of choice if “subcategory” and “equivalence” have their naive (‘strong’) meanings. For given a surjection $p:A\to B$ in $\mathrm{Set}$, make $A$ into a category with a unique isomorphism $a\cong a\prime$ iff $p(a)=p(a\prime )$; then a skeleton of $A$ supplies a splitting of $p$.

However, in the absence of choice, it is more appropriate to define a skeleton of $C$ to be any equivalent skeletal category, where now “equivalence” is understood in the sense of anafunctors or weak equivalence. In the presence of choice this definition is not much different, since any strong equivalence of categories $D\to C$, where $D$ is skeletal, exhibits $D$ as isomorphic (not merely equivalent) to a skeletal subcategory of $C$. This is no longer true for an ana-equivalence or weak equivalence. However, even with this more general notion of equivalence, some amount of choice is required to show that every category has a skeleton (although, for instance, any preorder has a skeleton in this sense without any need for choice).

Mike: It would be interesting to know the precise strength of the statement “every category is (ana-)equivalent to a skeletal one.”

Notice that the axiom of choice fails in general when one considers internal categories. Hence not every internal category has a skeleton.

Equivalents of choice

Define a coskeleton of a category $C$ to be a skeletal category $S$ with a surjective equivalence $C\to S$. In Categories, Allegories? it is shown that the following are equivalent.

1. Any two ana-equivalent categories are strongly equivalent.

   I removed ‘non-ana’, since I don't think that ‘strongly equivalent’ would ever be used in an ‘ana-’ sense. —Toby

   Addendum: Actually, I don't know why I asked whether you meant weakly or strongly here, since obviously one can prove that two ana-equivalent categories are weakly equivalent! It seems that the discussion above used the terms ‘equivalence’ and ‘ana-equivalence’ where equivalence of categories uses ‘strong equivalence’ and ‘weak equivalence’ or ‘ana-equivalence’; so I just changed it. And then I added another entry, which maybe you should remove if Freyd & Scedrov don't actually address it. On the other hand, if they really talk about weak equivalence instead of ana-equivalence (although if they define it in elementary terms, it's hard to tell the difference), maybe there's no need to say ‘ana-’ at all on this page.

2. Any two weakly equivalent categories are strongly equivalent.

3. Every small category has a skeleton.

4. Every small category has a coskeleton.

5. Any two skeletons of a given small category are isomorphic.

6. Any two coskeletons of a given small category are isomorphic.

For convenience we add:

1. Given an inhabited family $\{S_i{\}}_I$ of equinumerous sets there exists $0\in I$ and a family of isomorphisms of the permutation groups $\{\mathrm{Aut}(S_0)\to \mathrm{Aut}(S_i){\}}_I$.
2. Given a family $\{S_i{\}}_I$ of inhabited equinumerous sets, there exists a family $(x_i{)}_I$ such that $x_i\in S_i$ for all $i\in I$.

Uniqueness of constructions

It is well-known that objects defined by universal properties in a category, such as limits and colimits, are not unique on the nose, but only unique up to unique canonical isomorphism. It can be tempting to suppose that in a skeletal category, where any two isomorphic objects are equal, such objects will in fact be unique on the nose. However, under the most appropriate definition of “unique,” this is not true (in general), because of the presence of automorphisms.

More explicitly, consider the notion cartesian product in a category. Although we colloquially speak of “a product” of objects $A$ and $B$ as being the object $A\times B$, strictly speaking a product consists of the object $A\times B$ together with the projections $A\times B\to A$ and $A\times B\to B$ which exhibit its universal property. Thus, even if the category in question is skeletal, so that there can be only one object $A\times B$ that is a product of $A$ and $B$, in general this object can still “be the product of $A$ and $B$” in many different ways.

For example, given any projections $A\times B\to A$ and $A\times B\to B$ that exhibit $A\times B$ as a product of $A$ and $B$, we can compose them both with any automorphism of $A\times B$ to get a new, different, pair of projections that also exhibit $A\times B$ as a product of $A$ and $B$. In fact, the universal property of a product implies that any two pairs of projections are related by an automorphism of $A\times B$, so this example is generic. Thus, even in a skeletal category, we cannot speak of “the” product of $A$ and $B$, except in the same generalized sense that makes sense in any category. A formal way to say this is that the “category of products of $A$ and $B$,” while still equivalent to the trivial category, as it is in any category with products, will not be isomorphic to the trivial category even when the ambient category is skeletal.

(It is true in a few cases, though, that skeletality implies uniqueness on the nose. For instance, a terminal object can have no nonidentity automorphisms, so in a skeletal category, a terminal object (if one exists) really is unique on the nose.)