Why do Delta-sets not allow quotients? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T06:49:02Z http://mathoverflow.net/feeds/question/22855 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22855/why-do-delta-sets-not-allow-quotients Why do Delta-sets not allow quotients? ferret 2010-04-28T14:31:16Z 2010-04-28T15:06:12Z <p>A $\Delta$-set is a contravariant functor from the category $\Delta'$ of order-preserving injections to the category of sets (this is essentially what Allen Hatcher calls a $\Delta$-complex).</p> <p>A main reason for working with simplicial sets instead of $\Delta$-sets should be that they allow quotients (see e.g. Allen Hatcher's nice appendix "CW complexes with simplicial structure" to his Algebraic Topology book: "A major disadvantage of $\Delta$-complexes is that they do not allow quotient constructions"), How does this go well with the fact that the category of functors $\Delta'op\to Sets$ <strong>has</strong> colimits?</p> <p>(This question was already asked in a comment on Allen Hatcher's answer to <a href="http://mathoverflow.net/questions/6281/definition-of-simplicial-complex/6302#6302" rel="nofollow">this</a> question on the definition of simplicial complexes. I apologize for asking it twice but there has been no answer given and I am afraid that the reason is - if it's not the silliness of my question - that the comment appears only after pressing the "more comments" button. However, I apologize.)</p> http://mathoverflow.net/questions/22855/why-do-delta-sets-not-allow-quotients/22864#22864 Answer by Tyler Lawson for Why do Delta-sets not allow quotients? Tyler Lawson 2010-04-28T15:06:12Z 2010-04-28T15:06:12Z <p>The basic issue is that not every function that we would like to describe between $\Delta$-complexes can be realized by a natural transformation between functors. The lack of degeneracy maps means that no map $X \to Y$ of $\Delta$-complexes that sends any simplex down to a degenerate simplex can be realized by a natural transformation of functors. For example, if $X$ is a $\Delta$-complex interval realizing $[0,1]$ and $Y$ is a $\Delta$-complex realizing $[0,1]^2$, then there is no natural transformation of functors realizing the projection maps <code>$p_i:[0,1]^2 \to [0,1]$</code>.</p> <p>As a consequence, the category of $\Delta$-complexes does not have enough immediately-available maps between objects to construct the kinds of colimit diagrams one would like to realize.</p>