nLab skeleton

This entry is about the notion of skeleton in category theory. For the notion of (co)skeletal simplicial sets see at simplicial skeleton.

Definition

A strict category $\mathcal{C}$ is called skeletal if any two objects that are isomorphic are actually already equal (in the fixed set of objects $Obj(\mathcal{C})$), hence if all its isomorphisms are automorphisms.

(If in addition all isomorphisms are in fact identity morphisms, then one speaks of a gaunt category. In the other extreme, if all morphisms of a skeletal category are isomorphisms, then one has a skeletal groupoid).

A skeleton of a category $C$ is defined to be a skeletal subcategory of $C$ whose inclusion functor exhibits it as equivalent to $C$. A weak skeleton of $C$ is any skeletal category which is weakly equivalent to $C$.

In the absence of the axiom of choice, it is more appropriate to define a skeleton of $C$ to be a weak skeleton as defined above.

Constructions

In this section, we work in set-level foundations to construct skeleta of categories and more general objects.

Existence of Skeletons of Categories

Theorem

If the axiom of choice holds, then every category $C$ has a skeleton (in the strongest sense).

Proof

Simply choose one object in each isomorphism class and one isomorphism to that object from each other object in that class.

In more detail, generate the full subcategory $sk(C)$ containing just the chosen objects. Denote by $in\colon sk(C)\to C$ the inclusion. We exhibit a weak inverse of $in$ as a functor $-':x\mapsto x'$ constructed as follows. For every object $x$ one has chosen already the unique object $x'$ in $sk(X)$ isomorphic to $x$, but one also needs to make a choice of isomorphism $i_x\colon x\to x'$ for every $x$. This enables to conjugate between $C(x,y)$ and $C(x',y')$ by

$$(x\stackrel{f}\to y)\mapsto (x'\stackrel{i_x^{-1}}\to x\stackrel{f}\to y\stackrel{i_y}\to y').$$

The rule for morphisms $-': f\mapsto f' := i_y\circ f\circ i_{x}^{-1}$ is clearly functorial. Let us show that $-'$ is a weak inverse of $in$. In one direction, given $y\in sk(C)$ we compute $(in_{y})' = y$ (strict equality); in another direction, given $x\in C$, notice that $i^{-1}_{in_{x'}}:in_{x'}\cong x$ for $x\in C$ is an isomorphism. It suffices to show that these isomorphisms for all $x\in C$ together form a natural isomorphism $i^{-1}_{in}:in_{-'}\to id_C$; the naturality diagram is commutative precisely because of the conjugation formula for the functor $-'$ for morphisms. This completes the proof that $-'$ is indeed a weak inverse of $in$.

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 $Set$, make $A$ into a category with a unique isomorphism $a\cong a'$ iff $p(a)=p(a')$; then a skeleton of $A$ supplies a splitting of $p$.

Even with the more general notion of weak or ana-equivalence of categories, some amount of choice is required to show that every category has a skeleton. It would be interesting to know the precise strength of the statement “every category is weakly equivalent to a skeletal one.”

Skeletons as an Endo-Pseudofunctor on $\mathfrak{Cat}$

In the presence of choice we can thusly turn the process of taking the skeleton of a category into an endo-pseudofunctor on $\mathfrak{Cat}$, the $2$-category of categories.

For each category $\mathcal{C}$ let $sk_\mathcal{C}$ denote a skeleton of $\mathcal{C}$ and let $E_\mathcal{C}:sk_\mathcal{C}\simeq\mathcal{C}:E_\mathcal{C}^{\sim1}$ denote the skeletal equivalence at $\mathcal{C}$, with $\epsilon^{sk}_\mathcal{C}:E_\mathcal{C}\circ E^{\sim1}_\mathcal{C}\Rightarrow1_\mathcal{C}$ the skeletal transformation at $\mathcal{C}$. We define an endo-pseudofunctor

$$sk:\mathfrak{Cat}\to\mathfrak{Cat}$$

as follows:

1. $sk(\mathcal{C})=sk_\mathcal{C},$
2. $sk(F:\mathcal{C}\to\mathcal{D})=E^{\sim1}_\mathcal{D}\circ F\circ E_\mathcal{C}:sk_\mathcal{C}\to\mathcal{C}\to\mathcal{D}\to sk_\mathcal{D},$
3. $sk(\alpha:F\Rightarrow G)=i\circ\alpha\circ i^{-1}:sk(F)\Rightarrow sk(G),$ where
   $$i\circ\alpha\circ i^{-1}:sk(F)\Rightarrow sk(G)$$

   is defined by

   $$(i\circ\alpha\circ i^{-1})_X=i_{G(X)}\circ\alpha_X\circ i_{F(X)}^{-1}:F(X)'\to F(X)\to G(X)\to G(X)'.$$
4. For each category $\mathcal{C}$, the unitor $\iota^{sk}_\mathcal{C}:1_{sk(\mathcal{C})}\Rightarrow sk(1_\mathcal{C})$ is defined by
   $$\iota^{sk}_\mathcal{C}=1_{1_{sk(\mathcal{C})}},$$

   since

   $$sk(1_\mathcal{C})=E^{\sim1}_\mathcal{C}\circ1_\mathcal{C}\circ E_\mathcal{C}=E^{\sim1}_\mathcal{C}\circ E_\mathcal{C}=1_{sk(\mathcal{C})}.$$
5. For each pair of composable functors $F:\mathcal{C}\to\mathcal{D}$, $G:\mathcal{B}\to\mathcal{C}$, the associator $\alpha^{sk}_{F,G}:sk(F)\circ sk(G)\Rightarrow sk(F\circ G)$ is defined by
   $$\alpha^{sk}_{F,G}=1_{E^{\sim1}_\mathcal{D}\circ F}\star\epsilon^{sk}_\mathcal{C}\star1_{G\circ E_\mathcal{B}}:E^{\sim1}_\mathcal{D}\circ F\circ E_\mathcal{C}\circ E_\mathcal{C}^{\sim1}\circ G\circ E_\mathcal{B}\Rightarrow E^{\sim1}_\mathcal{D}\circ F\circ G\circ E_\mathcal{B}.$$

Skeleton of an Indexed Category

Using the above definition, we can canonically define the skeleton of an indexed category.

Let $\psi:\mathcal{B}^{op}\to\mathfrak{Cat}$ be an indexed category. We define the indexed skeleton of $\psi$, denoted

$$sk(\psi):\mathcal{B}^{op}\to\mathfrak{Cat},$$

to be the indexed category given by postcomposing $\psi$ with $sk$, so $sk(\psi)=sk\circ\psi.$ That is, for each $I\in\mathcal{B}$ we take a skeleton $sk(\psi(I))$ of the category $\psi(I)$ indexed by $I$ with equivalence $E_I:\psi(I)\simeq sk(\psi(I)):E_I^{\sim1}$, we extend functors using the equivalences at various skeletons, and we extend natural transformations using skeletal isomorphisms in the codomain categories (although $\mathcal{B}^{op}$ has only identity $2$-cells since it’s a promoted $1$-category, so the action on $2$-cells is trivial).

Skeleton of a Fibered Category

Using the above definition and the Grothendieck construction, we can define the skeleton of a fibration using the skeleton of an indexed category.

Let $p:\mathcal{E}\to\mathcal{B}$ be a fibration. We define the fibered skeleton of $p$, denoted

$$sk(p):sk^p_\mathcal{E}\to\mathcal{B}$$

to be the Grothendieck completion of the indexed skeleton of the indexing by $p$. That is, $sk(p)$ is the fibration obtained by turning $p$ into an indexed category using the Grothendieck construction, taking the indexed skeleton of this indexed category, then turning the resulting skeletal indexed category back into a fibration (via the Grothendieck construction again). We refer to the total category $sk^p_\mathcal{E}$ in the resulting fibration as the $p$-skeleton of $\mathcal{E}$, which is different than $sk_\mathcal{E}$ since we have only skeletalized the fiber categories of our original fibration and not the total category of the original fibration. In general, we will have that

$$sk_\mathcal{E}\ncong sk^p_\mathcal{E},$$

although they are trivially equivalent. As an example of this construction in action, we can pass from the monomorphism fibration on a category with pullbacks to the subobject fibration by taking the fibered skeleton. For details, see the subobject fibration section of codomain fibration.

Properties

In this section we collect some properties of skeleta in set-level foundations.

Non-invariance

Of course, two categories can be equivalent even if one is skeletal and the other is not. Thus, the notion of skeletal category violates the principle of equivalence. (It doesn’t make sense to ask whether a particular category, skeletal or not, “violates the principle of equivalence”.)

Skeletons without Choice

Without any choice, we have the following theorems.

Theorem

Any thin category (i.e. any preordered set) has a weak skeleton.

Proof

In this case, we can take the objects of the skeleton of $C$ to be the isomorphism classes of $C$. If $C$ is thin, then we can define a partial ordering on its set of isomorphism classes, making them into a skeleton of $C$.

Theorem

Any two skeletons (in the strong sense) of a category are isomorphic.

Proof

Let $\mathcal{C}$ be a category, with $sk (\mathcal{C})$ and $sk (\mathcal{C})'$ two skeletons of $\mathcal{C}$, and for each object $Y\in\mathcal{C}$ denote by $X_Y$ and $X_Y'$ respectively the unique objects in $sk (\mathcal{C})$ and $sk (\mathcal{C})'$ which are isomorphic to Y, and denote their respective isomorphisms by $i_Y:Y\to X_Y$ and $i_Y':Y\to X_Y'$. We define a functor $F:sk (\mathcal{C})\to sk (\mathcal{C})'$ by

$$F(X_Y)=X_Y',$$

$$F(f:X_Y\to X_Z)=i_Z'\circ i_Z^{-1}\circ f\circ i_Y\circ i_Y'^{-1}:X_Y'\to Y\to X_Y\to X_Z\to Z\to X_Z'.$$

We then have

$$F(1_{X_Y})=i_Y'\circ i_Y^{-1}\circ 1_{X_Y}\circ i_Y\circ i_Y'^{-1}=i_Y'\circ i_Y^{-1}\circ i_Y\circ i_Y'^{-1}=i_Y'\circ1_Y\circ i_Y'^{-1}=i_Y'\circ i_Y'^{-1}=1_{X_Y'}=1_{F(X_Y)},$$

$$F(f:X_B\to X_C)\circ F(g:X_A\to X_B)=i_C'\circ i_C^{-1}\circ f\circ i_B\circ i_B'^{-1}\circ i_B'\circ i_B^{-1}\circ g\circ i_A\circ i_A'^{-1}$$

$$=i_C'\circ i_C^{-1}\circ f\circ i_B\circ 1_{B}\circ i_B^{-1}\circ g\circ i_A\circ i_A'^{-1}=i_C'\circ i_C^{-1}\circ f\circ i_B\circ i_B^{-1}\circ g\circ i_A\circ i_A'^{-1}$$

$$=i_C'\circ i_C^{-1}\circ f\circ1_{X_B}\circ g\circ i_A\circ i_A'^{-1}=i_C'\circ i_C^{-1}\circ f\circ g\circ i_A\circ i_A'^{-1}=F(f\circ g),$$

so $F$ is a functor. The inverse $F^{-1}: sk (\mathcal{C})'\to sk (\mathcal{C})$ is defined in the obvious way, and is functorial by similar computations.

Notice that the axiom of choice fails in general when one considers internal categories. Hence not every internal category has a skeleton. A necessary condition for an internal category $X_1 \rightrightarrows X_0$ to have a skeleton is the existence quotient $X_0/X_1$ - the object of orbits under the action of the core of $X$. If the quotient map $X_0 \to X_0/X_1$ has a section, then one could consider $X$ to have a skeleton, but this condition isn’t sufficient for the induced inclusion functor to be a weak equivalence of internal categories when this makes sense (i.e. if the category is internal to a site).

David Roberts: The claim above about the necessity of the existence of the quotient needs to be checked.

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. The axiom of choice holds.
2. Any two ana-equivalent categories are strongly equivalent.
3. Any two weakly equivalent categories are strongly equivalent.
4. Every small category has a weak skeleton.
5. Every small category has a coskeleton.
6. Any two weak skeletons of a given small category are isomorphic.
7. 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 $\{Aut(S_0) \to Aut(S_i)\}_I$.
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 of 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.)

In intensional/homotopy type theory

All the discussion above pertains to set-level foundations. In intensional type theory such as homotopy type theory, additional care is needed.

A first guess is to define a precategory $C$ to be skeletal if for any objects $a,b$ there is a function $\Vert a\cong b\Vert \to (a=b)$, i.e. if $a$ and $b$ are merely isomorphic then they are equal. (Here $\Vert-\Vert$ denotes the propositional truncation.) However, if the type $ob(C)$ of objects is not a set, then $a=b$ is not a proposition, and so under this definition a category could “be skeletal” in more than one way.

Arguably a better definition is to say that a precategory $C$ is skeletal if for any objects $a,b$ the canonical function

$$(a=b) \to \Vert a\cong b\Vert$$

is an equivalence. Since $\Vert a\cong b\Vert$ is always a proposition, this entails that $a=b$ is also a proposition. Thus, this implies that $ob(C)$ is a set, i.e. that $C$ is a strict category.

Note the analogy between this condition and the notion of univalent category: the only difference is the propositional truncation. It follows that if a precategory is both skeletal and univalent, then $a\cong b$ is always a proposition (since it is equivalent to its propositional truncation), and thus the category is gaunt. Indeed, the gaunt categories are precisely those that are both skeletal and univalent.

Unlike in set-level foundations, in intensional type theory not every precategory (and not even every univalent category) has a skeleton. Indeed, if the axiom sets cover fails, then there can be univalent categories that are not weakly equivalent to any strict category. For this reason, the notion of skeleton tends to be less useful outside of set-level foundations, although particular concrete examples may turn out to be skeletal.

Examples

• Every poset is skeletal.

• The walking parallel pair is a skeletal category which is not a poset.

• The free category on a directed acyclic graph? is skeletal.

• The category of Archimedean ordered fields and monotonic field homomorphisms is skeletal.

All of the above examples are in fact gaunt: not only are any two isomorphic objects equal, but any isomorphism is an identity morphism.

• The delooping of a group is a skeletal category that is not gaunt (unless the group is trivial). In intensional type theory, this is a non-univalent precategory; its Rezk completion is no longer skeletal according to the above improved definition.

Posets as skeleta of prosets

In terms of (n,r)-category-theory one may essentially identify preordered sets with thin categories or (0,1)-categories. Under this identification, the passage of skeleta of categories corresponds to choosing an equivalent partial order inside a preorder.

For more details (and more precise statements, see at relation between preorders and (0,1)-categories.

Related concepts

• skeletal groupoid

References

• Saunders MacLane, p. 91 of: Categories for the Working Mathematician, Graduate texts in mathematics, Springer (1971) [doi:10.1007/978-1-4757-4721-8]

• Emily Riehl, p. 34 in: Category Theory in Context, Dover Publications (2017) [pdf]

• Birgit Richter, §2.6 in: From categories to homotopy theory, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press (2020) [doi:10.1017/9781108855891, book webpage, pdf]

See also:

• Wikipedia, Skeleton_(category_theory)

Presenting a pretorsion theory on Cat whose torsion(-free) objects are the groupoids (skeletal categories, respectively), hence whose “trivial objects” are the skeletal groupoids:

• Francis Borceux, Federico Campanini, Marino Gran, Walter Tholen, Groupoids and skeletal categories form a pretorsion theory in $Cat$ [arXiv:2207.08487]