(Homotopy theory) When does the 2 of 3 property not imply 2 of 6? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:48:35Z http://mathoverflow.net/feeds/question/120872 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120872/homotopy-theory-when-does-the-2-of-3-property-not-imply-2-of-6 (Homotopy theory) When does the 2 of 3 property not imply 2 of 6? Jesse Wolfson 2013-02-05T16:14:52Z 2013-02-05T23:49:10Z <p>A relative category is a category $C$ with a subcategory $W$ containing all the objects of $C$. </p> <p>Given a relative category $(C,W)$, $W$ is said to satisfy the 2 implies 6'' property if, for any collection of three composable maps,</p> <p>$$X\rightarrow Y\rightarrow Z\rightarrow A$$</p> <p>the presence of the composites $X\rightarrow Z$ and $Y\rightarrow A$ in $W$ implies that each individual map is in $W$ (and so also the triple composition).</p> <p>The property I'm more familiar with from thinking about weak equivalences is the 2 implies 3" property, which says that, given a pair of composable maps</p> <p>$$X\rightarrow Y\rightarrow Z$$</p> <p>the presence of any two of the maps </p> <p>$$X\rightarrow Y$$ $$Y\rightarrow Z$$ $$X\rightarrow Z$$</p> <p>in $W$ implies that the third is as well.</p> <p>The "2 implies 6" property implies the "2 implies 3" property, and I've been told that "2 implies 6" is a strictly stronger property. </p> <p>QUESTION: What is the basic example of a relative category $(C,W)$ where $W$ satisfies "2 implies 3", but not "2 implies 6"? </p> <p>Edit: By "basic", I mean what is an example which comes up in applications, or better yet, what is the example to keep in mind?</p> http://mathoverflow.net/questions/120872/homotopy-theory-when-does-the-2-of-3-property-not-imply-2-of-6/120874#120874 Answer by Simon Rose for (Homotopy theory) When does the 2 of 3 property not imply 2 of 6? Simon Rose 2013-02-05T16:33:53Z 2013-02-05T16:47:51Z <p><strong>Edit:</strong> This answer only makes sense if the actual question was about morphism $X \to A$ and not $X \to Z$.</p> <p>Couldn't you just pick a category with two objects $x, y$ with only one invertible morphism $f$ between the two of them? Then consider $$x \to x \to y \to x$$ with the morphism $x \to x$ being the identity. If we let $W$ be the subcategory consisting of $x, y$ but not the morphism between them, then it satisfies your condition that $X \to Y$ and $Y \to A$ (both of which are just the identity $x \to x$), but not that $x \to y$ or $y \to x$ are in $W$.</p> <p>In this case, $W$ satisfies the 2-implies-3, since if the diagrams $$X \to Y \to Z$$ have 2 morphisms that are in $W$, then they must be of the form $x \to x \to x$ or $y \to y \to y$, and so it satisfies the 2-implies-3 condition.</p> http://mathoverflow.net/questions/120872/homotopy-theory-when-does-the-2-of-3-property-not-imply-2-of-6/120878#120878 Answer by Eric Wofsey for (Homotopy theory) When does the 2 of 3 property not imply 2 of 6? Eric Wofsey 2013-02-05T16:47:11Z 2013-02-05T16:47:11Z <p>Here's a rather tautological example. Consider the category $$X\rightarrow Y\rightarrow Z\rightarrow A.$$ That is, $X$, $Y$, $Z$, and $A$ are the only objects, and the only morphisms are those appearing in the diagram (and their composites). Then let $W$ consist of the identity maps, the map $X\to Z$, and the map $Y\to A$. Then this satisfies 2 out of 3 but not 2 out of 6.</p> http://mathoverflow.net/questions/120872/homotopy-theory-when-does-the-2-of-3-property-not-imply-2-of-6/120917#120917 Answer by Tom Goodwillie for (Homotopy theory) When does the 2 of 3 property not imply 2 of 6? Tom Goodwillie 2013-02-05T23:49:10Z 2013-02-05T23:49:10Z <p>I'm not sure that any examples naturally come up, of cases where you have the 2 out of 3 condition but not the 2 out of 6. Of course, if membership in W is defined by requiring certain functors to take a morphism to isomorphisms (as is so often the case in applications), then you always have 2 out of 6 (because a morphism that has both a left inverse and a right inverse always has an inverse). </p> <p>In Quillen's model category axioms 2 out of 3 is an axiom and 2 out of 6 follows from this and the other axioms.</p>