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916823d | 1 2 3 4 5 6 7 8 9 10 11 12 | Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$.
A $\textbf{universal arrow}$ over $x\in \mathcal B$ is a structured arrow $u:x\to U(c)$ with domain $x$ that has the following universal property: for each structured arrow $f: x\to U(b)$ with domain $x$ there exists a unique morphism $\underline{f} : c\to b$ such that $\underline f\circ u=f$.
Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$.
A $\textbf{free object}$ over $x\in \mathcal B$ is an object $c\in \mathcal C$ such that there exists a universal arrow $(u,c)$ over $x$.
Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$.
$\mathcal C$ is said to $\textbf{have free objects}$, if for each $x\in \mathcal B$ there is a free object over $x$.
Theorem: Let $\mathcal{L}\mathrm{at}_{\lor}^{\infty}$ be the category of suplattices.
The consturct $\mathcal{L}\mathrm{at}_{\lor}^{\infty}$ has free objects. |