when are epimorphisms of algebraic objects surjective? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:50:45Z http://mathoverflow.net/feeds/question/10231 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10231/when-are-epimorphisms-of-algebraic-objects-surjective when are epimorphisms of algebraic objects surjective? Martin Brandenburg 2009-12-31T04:50:03Z 2010-01-04T18:42:24Z <p>let $C$ be the category of $\tau$-algebras for some type $\tau$. consider the statements:</p> <ol> <li>every monomorphism is regular.</li> <li>every epimorphism in C is surjective.</li> </ol> <p>it is easy to see that 1. implies 2. what about the converse?</p> http://mathoverflow.net/questions/10231/when-are-epimorphisms-of-algebraic-objects-surjective/10372#10372 Answer by Todd Trimble for when are epimorphisms of algebraic objects surjective? Todd Trimble 2010-01-01T13:56:39Z 2010-01-01T13:56:39Z <p>Certainly not. Consider the category of groups: every epimorphism is surjective (see, e.g., Categories for the Working Mathematician, p. 21 exercise 5) but not every mono is a kernel. </p> http://mathoverflow.net/questions/10231/when-are-epimorphisms-of-algebraic-objects-surjective/10553#10553 Answer by Todd Trimble for when are epimorphisms of algebraic objects surjective? Todd Trimble 2010-01-03T01:07:08Z 2010-01-04T18:42:24Z <p>Update: the following exchange appeared on the categories mailing list several years ago: <a href="http://article.gmane.org/gmane.science.mathematics.categories/3094" rel="nofollow">http://article.gmane.org/gmane.science.mathematics.categories/3094</a>. Walter Tholen's response strongly suggests that the answer to Martin's question is that the converse does not hold, although I don't have access to the four-author article he cites as reference. It's probably worth a look though, and if I learn anything more I'll post another update. </p> <p><b>Second update</b>: My surmise was correct. Walter Tholen kindly emailed to me the relevant two pages (pp. 88-89) of the four-author paper </p> <ul> <li> E.W. Kiss, L. Marki, P. Prohle, W. Tholen: Studia Sci. Math. Hungaricum 18 (1983) 79-141. </li> </ul> <p>where the following example is given on page 89: in the category of semigroups with zero such that all 4-fold products are zero, all epimorphisms are surjective but not all monos are regular [a specific nonregular mono is described]. (I can forward this email if you write to me at topological dot musings at gmail dot com.) </p> <p>Assuming amalgamated products exist (as they do in categories of algebras of a Lawvere theory), a mono i: A >--> B is regular if it is the equalizer of the pair of canonical maps from B to the amalgamated product B *_A B (i.e., the coprojections of the pushout of i with itself, aka the <i>cokernel pair</i> of i). The equalizer of the cokernel pair defines a closure operator on the lattice of subalgebras Sub(B), called the <i>dominion operator</i> Dom_B. So to prove a subalgebra is not regular is to show that it is not Dom-closed. The key technical result needed to prove the claim above is Isbell's Zig-Zag theorem (given in his paper Epimorphisms and Dominions in the 1965 La Jolla conference proceedings on categorical algebra), as recalled <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.pjm/1102645825" rel="nofollow">here</a>, which gives a precise and useful criterion for an element to belong to the dominion (= Dom-closure) of a subalgebra. </p> <p>Hope this helps. I am voting up your question, Martin, since it's rather nontrivial! </p>