Initial and terminal objects

Initial and terminal objects are objects within a category which, if they exist, are unique up to isomorphism. Loosely speaking, all objects ‘flow’ from the initial object and to the terminal object. More precisely, in a category $𝖢$ , objects $𝟎$ and $𝟏$ are called the initial and terminal objects respectively if for any object $𝑋$ there exist unique morphisms $𝐼 \in 𝖢 (𝟎, 𝑋)$ and $𝑇 \in 𝖢 (𝑋, 𝟏)$ . cat

𝟎 𝐼 \leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow \to 𝑋 𝑇 \leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow \to 𝟏

Concisely, $𝖢 (𝟎, -)$ and $𝖢 (-, 𝟏)$ always contain exactly one morphism.

Uniqueness up to isomorphism

Let $𝟎'$ be an object in $𝖢$ with the initial property. Then there exists unique $𝑓 \in 𝖢 (𝟎, 𝟎')$ and $𝑔 \in 𝖢 (𝟎', 𝟎)$ . Likewise the only endomorphisms are $i d 𝟎$ and $i d 𝟎'$ . Hence $𝑔 𝑓 = i d 𝟎$ and $𝑓 𝑔 = i d 𝟎'$ , therefore $𝟎 ≅ 𝟎'$ .

Likewise let $𝟏'$ be an object in $𝖢$ with the initial property. Then there exists unique $𝑓 \in 𝖢 (𝟏, 𝟏')$ and $𝑔 \in 𝖢 (𝟏', 𝟏)$ . Likewise the only endomorphisms are $i d 𝟎$ and $i d 𝟎'$ . Hence $𝑔 𝑓 = i d 𝟏$ and $𝑓 𝑔 = i d 𝟏'$ , therefore $𝟏 ≅ 𝟏'$ .

As limits and colimits

Formulated as Limits and colimits, the terminal object is the limit of the empty diagram and the initial object is its colimit.

𝟏 : = l i m ⟵ \emptyset 𝟎 : = l i m ⟶ \emptyset

Examples

In a Poset

The simplest example is perhaps in posets, viewed as categories, in which the initial and terminal objects represent the smallest and largest values respectively.

In Category of sets and Category of topological spaces

In the category \Set, it is required that a unique morphism exists mapping the empty set $\emptyset$ for every set $𝐴$ . Hence $\emptyset$ is the initial object.

In a similar fashion, it is clear that one and only one mapping exists from each set to a singleton set ${*}$ , and that all singletons are isomorphic. Hence the singleton is the terminal object.

In Hask

Analogously, in $𝖧 𝖺 𝗌 𝗄$ , the initial object is the Void type

absurd :: Void -> a

While the terminal object is the canonical singleton type ()

unit :: a -> ()
unit _ = ()

In Category of vector spaces

In $𝖵 𝖾 𝖼 𝗍 𝕂$ , both the initial and terminal object, hence the zero object, is the trivial vector space $𝑂$ . Clearly, all trivial vector spaces are isomorphic (e.g. the trivial subspace of $ℝ 2$ and $ℝ 3$ ). For any vector space $𝑉 \in O b (𝖵 𝖾 𝖼 𝗍 𝕂)$ , there exists exactly one linear transformation $𝑓 \in 𝖵 𝖾 𝖼 𝗍 𝕂 (𝑉, {⃗ 𝟎})$ , and this is also clearly epic.

𝑓 : 𝑉 ↣ 𝑂 : 𝑣 \mapsto ⃗ 𝟎

Likewise, there exists exactly one (monic) linear transformation $𝑔 \in 𝖵 𝖾 𝖼 𝗍 𝕂 ({⃗ 𝟎}, 𝑉)$

𝑔 : 𝑂 ↠ 𝑉 : ⃗ 𝟎 \to ⃗ 𝟎

and we therefore have $𝑓 \circ 𝑔 = i d 𝑂$

In Category of groups

The trivial group ${𝑒}$ is both the initial and terminal object.

tidy | SemBr | en

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