## § Category where coproducts of computable things is not computable

• Modular lattices are an algebraic variety.
• Consider the category of modular latties.
• The free modular lattice on 2 elements and on 3 elements has dediable equality, by virtue of being finite.
• The free modular lattice on 5 elements does not have decidable equality.
• The coproduct of free modular lattice on 2 and 3 generators is the free modular lattice on 5 generators, because $F(2 \cup 3) = F(2) \sqcup F(3)$ (where $2, 3$ are two and three element sets), because free is left adjoint to forgetful, and the left adjoint $F$ preserve colimits!