An terminal object has a unique morphism from every other object in the category.
Terminal objects give a category theory version of the concept of 'element' in set theory. 1 -> A allows us to pick out an arbitrary element of the set A.
Terminal Object in SetAlthough category theory is about the external properties of objects, to get an intuitive understanding, it can be helpful to peek inside them. In set-like categories the terminal object is like the single element set. In set, maps exist from any n-element to an m-element set (provided m≠0). So every set has maps to every non-empty set, however only maps to the 1-element set are unique. |
Examples in Various Categories
| Terminal Object | ||
|---|---|---|
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| Notation | 1 | |
| generalisation | a kind of limit | |
| universal cone over diagram |
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| examples: set: | {1}or {a} ... set with one element (singleton) |
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| group (null object) | trivial group (just identity element) | |
| topological space | single point | |
| poset | greatest element (if exists) | |
| monoid | trivial monoid (consisting of only the identity element) | |
| semigroup | singleton semigroup | |
| Rng | trivial ring consisting only of a single element 0=1 | |
| fields | does not have terminal object | |
| Vec | zero object | |
| Top | one-point space | |
| Grf | graph with a single vertex and a single loop | |
| Ω-Alg algebra with signature Ω |
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| Cat | category 1 (with a single object and morphism) | |
empty diagram