nLab rigid object

Definition

An object $x$ in a category $C$ is said to be rigid if its automorphism group is trivial, in other words if any isomorphism $f : x \cong x$ must be equal to the identity morphism $f = 1_x : x \to x$.

More generally, an object of an n-category (or (n,r)-category, etc.) is rigid if its higher category $Aut(x)$ of automorphisms is terminal.

Idea

Certain types of questions are made much more difficult to answer by the presence of non-trivial automorphisms (or “symmetries”). In such settings, sometimes it is helpful to first consider the question for rigid objects, and then try to extend the answer to non-rigid objects. It can even be useful to artificially “point” or “root” the objects of the original category $C$ by moving to a larger category $C_\bullet$ (with a forgetful functor $C_\bullet \to C$) in which all objects are rigid.

Examples

• As trivial examples, any initial object or terminal object is rigid, as is every object of a poset. (While trivial, such examples are also significant; e.g., universal properties may be formulated in terms of initial or terminal objects in suitable categories.)

• In graph theory, a rigid object in the category of undirected graphs is called an asymmetric graph.

• Let $C$ be the category of transitive G-sets for some group $G$. Although a transitive $G$-set may in general have non-trivial automorphisms, these symmetries can be killed off by moving to the larger category $C_\bullet$ of pointed transitive $G$-sets, whose objects are pairs of a transitive $G$-set $X$ equipped with an element $r \in X$, and whose morphisms $f : (X,r) \to (Y,s)$ are $G$-equivariant functions $f : X \to Y$ preserving the point $f(r) = s$. Indeed, suppose $f : (X,r) \to (X,r)$ is any endomorphism of $(X,r)$, and let $x \in X$ be any element. By the assumption that $G$ acts transitively on $X$, there exists $g \in G$ such that $x = g * r$. But then by equivariance and preservation of the point we have that

  $$f(x) = f(g*r) = g*f(r) = g*r = x.$$

  Note that this example is relevant to the combinatorics of embedded graphs (see at combinatorial map).

• In point-set topology, there are constructions of continua whose only continuous endomorphisms are constant maps and the identity. Examples include so-called “Cook continua”. Note that the interest is not merely in the pleasure of concocting exotic and pathological spaces: there is also some import for category theory, for instance in better understanding the problem of characterizing reflective subcategories of Top. See Kannan and Rajagopolan (and references therein) for some discussion.

• In the (∞,1)-category Grpd, an Eilenberg-MacLane space $K(G,1)$ is rigid if $G$ has trivial center and also trivial outer automorphism group, since $Aut(K(G,1))$ is a homotopy 1-type with $\pi_0(Aut(K(G,1)))=Out(G)$ and $\pi_1(Aut(K(G,1)))=Z(G)$. In particular, this is the case for $G=S_n$ the symmetric group on $n$ letters, where $n\ge 3$ and $n\neq 6$. This allows us to construct an embedding of $\mathbb{N}$ into the object classifier.

• A real closed field is a rigid object in the category Field of fields.

Related concepts

• groupoid cardinality

References

For a discussion of the example of pointed transitive $G$-sets (among other things), see

• Qiaochu Yuan, “The categorical exponential formula”, blog post at Annoying Precision, November 4, 2015. (web)

Rigid topological spaces are discussed in

• V. Kannan and M. Rajagopolan, Constructions and applications of rigid spaces, I, Adv. Math., Vol. 29 Iss. 1 (July 1978), 89-130. (web)