Does the category Monoid of monoids have finite coproducts? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:24:31Z http://mathoverflow.net/feeds/question/17965 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17965/does-the-category-monoid-of-monoids-have-finite-coproducts Does the category Monoid of monoids have finite coproducts? guy 2010-03-12T10:18:39Z 2010-03-13T19:03:56Z <p>Does the category Monoid of monoids have finite coproducts?</p> http://mathoverflow.net/questions/17965/does-the-category-monoid-of-monoids-have-finite-coproducts/17973#17973 Answer by Tom Leinster for Does the category Monoid of monoids have finite coproducts? Tom Leinster 2010-03-12T11:25:51Z 2010-03-12T11:25:51Z <p>Yes. More generally, any category of algebras (in the sense of universal algebra), such as groups, rings, vector space, Lie algebras, ..., has all (small) limits and colimits. See for instance p.210 (end of section IX.1) of Mac Lane's book <em>Categories for the Working Mathematician</em>.</p> <p>Explicitly, the initial monoid ("0-fold coproduct of monoids") is the one-element monoid. The coproduct \$A * B\$ of two monoids \$A, B\$ is constructed similarly to the coproduct of two groups (often called their "free product"). That is, it's the free monoid generated by all the elements of \$A\$ together with all the elements of \$B\$, quotiented out by all the relations that hold in \$A\$ and all the relations that hold in \$B\$.</p>