example. direct limit [rosiak2022sheaf, example 68] [tt-004X]

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Direct limits is the colimit of a diagram indexed by the ordinal category \(\omega \). In other words, for a diagram \[ X_1 \to X_2 \to X_3 \to X_4 \to \cdots \] its colimit is the direct limit \(\lim \limits _{\to } X_n\), defining a diagram of shape \(\omega +1\) :

Observe then that the colimit of a sequence of sets with the inclusions \[ X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \] recovers their union \(\bigcup \limits _{n \geq 0} X_n\).