Free category

Free categories are the free objects in Category of small categories, cat forming the left adjoint to the forgetful functor $𝑈 : 𝖢 𝖺 𝗍 \to 𝖰 𝗎 𝗂 𝗏$ to the Underlying quiver

𝐶 ⊣ 𝑈 : 𝖢 𝖺 𝗍 \to 𝖰 𝗎 𝗂 𝗏

The free category $𝐶 Γ = Γ ――$ is constructed by considering all composable words, called paths, as morphisms.

Universal property

If $𝖣$ is a Small category with Underlying quiver $𝑈 𝖣$ and $𝑓 \in 𝖰 𝗎 𝗂 𝗏 (Γ, 𝑈 𝖣)$ is a quiver homomorphism then there exists a unique adjunct $𝑔 \in 𝖢 𝖺 𝗍 (𝐶 Γ, 𝖣)$ such that the following diagram commutes:

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Free category

Universal property

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