- Any complete meet-semilattice that does not have joins. For example, the semilattice of subsets of $\mathbb{N}$ of size at most $n$ for a fixed $n$.
- Take any object $A$ in a category with all products and consider the full subcategory whose objects are all powers of $A$. This will usually fail to have coproducts. For example, the category of all sets whose cardinalities are powers of $2$.
Alex Kruckman
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