Subobject

A subgroup of an object in a category $𝖢$ generalizes the idea of subset, subgroup, vector subspace, and more. Formally, a subobject of an object $𝑋 \in 𝖢$ is an isomorphism class of monomorphisms over $𝑋$ . #m/def/cat For example, given the commuting diagram

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the monomorphisms $𝑠 1$ and $𝑠 2$ give the same subobject.

Preorder of subobjects

Subobjects of $𝑋$ are isomorphism classes in the preorder or thin category $𝖲 𝗎 𝖻 𝑋$ given by the Fibre category of $𝖲 𝗎 𝖻 ⇥ 𝖢$ .

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