Definition
In category theory, an object $S$ of a category $\mathcal{C}$ is called subterminal if for every object $X$ of $\mathcal{C}$ there exists at most one morphism $X \to S$. Equivalently, the hom‑set $\mathrm{Hom}_{\mathcal{C}}(X,S)$ is either empty or a singleton for all $X$. When $\mathcal{C}$ possesses a terminal object $1$, an object $S$ is subterminal precisely when it can be exhibited as a subobject of $1$, i.e. there exists a monomorphism $m\colon S \hookrightarrow 1$.
Basic properties
- Terminal objects are subterminal. A terminal object $1$ satisfies the defining condition with a unique morphism from any $X$ to $1$; thus it is a (maximal) subterminal object.
- Stability under pullback. If $S$ is subterminal and $f\colon Y \to S$ is any morphism, then the pullback of $f$ along any morphism exists and yields a subterminal object over $Y$.
- Products. Finite products of subterminal objects are again subterminal. In particular, the binary product of two subterminal objects, when it exists, coincides with their meet in the lattice of subobjects of a terminal object.
- Relation to the subobject classifier. In a topos, subterminal objects correspond bijectively to global elements of the subobject classifier $\Omega$; they represent truth‑values that are either “false” (the initial object) or “true” (the terminal object) together with any intermediate “partial truth” given by subobjects of $1$.
Examples
| Category | Subterminal objects |
|---|---|
| Set (sets and functions) | Sets with at most one element: the empty set $\varnothing$ and any singleton ${*}$. |
| Poset (partially ordered sets viewed as categories) | Elements that are below at most one element; equivalently, lower bounds of at most one element. |
| Top (topological spaces) | Spaces that have at most one continuous map from any space, which are precisely the empty space and any one‑point space. |
| Grp (groups) | The trivial group (terminal object) and the empty group does not exist; thus the only subterminal object is the trivial group. |
| Presheaf categories $[\mathcal{D}^{op},\mathbf{Set}]$ | Subterminal objects are subpresheaves of the terminal presheaf; concretely, they assign to each object of $\mathcal{D}$ either the empty set or a singleton, functorially. |
Relationships to other notions
- The dual concept is a subinitial object: an object $I$ such that there is at most one morphism $I \to X$ for every $X$.
- In a regular category, subterminal objects are precisely the regular subobjects of the terminal object.
- The collection of subterminal objects in a category with a terminal object forms a meet‑semilattice under the product (which corresponds to intersection of subobjects of $1$).
References
- S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, 1998 – Section IV.1.
- P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, vol. 1, Oxford University Press, 2002 – §1.5.
- R. Goldblatt, Topoi: The Categorial Analysis of Logic, Dover Publications, 2006 – Chapter 3.
See also
- Terminal object
- Subobject classifier
- Subinitial object
- Pullback
- Regular category
This entry summarises the standard categorical notion of a subterminal object as found in the literature.