2-groups are to crossed modules as 2-categories are to...? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T12:44:56Z http://mathoverflow.net/feeds/question/35622 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35622/2-groups-are-to-crossed-modules-as-2-categories-are-to 2-groups are to crossed modules as 2-categories are to...? EricForgy 2010-08-15T02:36:53Z 2010-08-16T16:41:29Z <p>Given a <a href="http://ncatlab.org/nlab/show/2-group" rel="nofollow">2-group</a> $\mathcal{G}$, you can construct a <a href="http://ncatlab.org/nlab/show/crossed%20module" rel="nofollow">crossed module</a> $(G,H,t,\alpha)$ and vice versa.</p> <p>Is there something similar you can say for <a href="http://ncatlab.org/nlab/show/strict%202-category" rel="nofollow">strict 2-categories</a>?</p> <p>In a personal attempt to understand strict 2-categories, I ended up constructing a speculative conceptual tool (whose validity remains to be seen) that I call the boundary of a 2-morphism. I've written up some raw notes here:</p> <ul> <li><a href="http://ncatlab.org/ericforgy/show/Boundary+of+a+2-Morphism+and+the+Interchange+Law" rel="nofollow">Boundary of a 2-Morphism and the Interchange Law</a></li> </ul> <p>The basic idea is that given morphisms $f,g:x\to y$ and a 2-morphism $\alpha:f\Rightarrow g$, we define its boundary as an endomorphism</p> <p>$$\partial\alpha:y\to y$$</p> <p>satisfying</p> <p>$$\partial\alpha\circ f = g.$$</p> <p>When the source of the 2-morphism is an identity morphism, then we have</p> <p>$$\partial\alpha = t(\alpha),$$</p> <p>which seems to relate things well to cross modules when all morphisms are invertible.</p> <p>I'm curious if there is anything like a crossed module, but where we're not dealing with groups and morphisms are not invertible. What I'm trying to cook up seems like it might be related to such a thing if it exists.</p> <p>Any thoughts and/or any comments on my notes would be greatly appreciated.</p> <p>PS: Apologies in advance if my writing is not very clear. I'm not a mathematician, but am trying to teach myself some basic higher category theory.</p> http://mathoverflow.net/questions/35622/2-groups-are-to-crossed-modules-as-2-categories-are-to/35725#35725 Answer by Theo Johnson-Freyd for 2-groups are to crossed modules as 2-categories are to...? Theo Johnson-Freyd 2010-08-16T04:56:33Z 2010-08-16T04:56:33Z <p>This is not an answer, but rather a "no go" observation. I claim that you should not expect 2-categories in general to have "crossed-module" like descriptions, or at least not any such description that's any easier to think about than "2-category". Part of what makes 2-groups easy is that they have lots and lots of symmetry. Ignoring the 2-morphisms (and 2-composition), the 1-morphisms form a group, so by group translation you can relate the structure between any two 1-morphisms to the structure between some 1-morphism and the identity. And that structure is group- or torsor-like, since if you ignore the 0-morphism and the 1-composision, the 1-morphisms are a groupoid.</p> <p>I expect that you can construct something for a 2-category with (1) only one 0-morphism and (2) all 1-morphisms invertible. I.e. this is a 2-group but relaxing the invertibility condition on arbitrary 2-morphisms. Then I would expect that this should correspond to a "crossed module of groups" where the second "group" $H$ need only be a monoid, although I haven't thought about the details.</p> http://mathoverflow.net/questions/35622/2-groups-are-to-crossed-modules-as-2-categories-are-to/35764#35764 Answer by Chris Schommer-Pries for 2-groups are to crossed modules as 2-categories are to...? Chris Schommer-Pries 2010-08-16T13:50:11Z 2010-08-16T16:41:29Z <p>I think Theo's "no go" is exactly right. Here is an example which might make things easier to understand: Let X be any category. I am going to construct an interesting 2-category with one object which is like a 2-group, but without the invertibility. So there is a single 0-morphism p. The morphisms from p to itself form a category which is a disjoint union of X and two points: $$0 \sqcup X \sqcup \infty$$ This is the disjoint union of categories so X and these other points don't interact. That completely describes the vertical composition. Now I need to tell you the horizontal composition. The element 0 is the (strict) identity for the horizontal composition. The point $\infty$ has the property that $z \cdot \infty = \infty = \infty \cdot z$ for any z. Finally the horizontal composite of any two things in X results in $\infty$. </p> <p>Equivalently we can describe this as a monoidal structure on $0 \sqcup X \sqcup \infty$. It is actually strictly commutative too. </p> <p>The reason this an important example is that we have embedded the category X fully-faithfully into this monoidal category. So any sort of algebraic description of monoidal categories or 2-categories or even strict 2-categories must be at least as complicated as the theory of <em>all categories</em>. This is in severe contrast with the situation for 2-groups for the reasons that Theo pointed out. </p> <p>This example is also related to Reid Barton's answer to my question: <a href="http://mathoverflow.net/questions/430/homological-algebra-for-commutative-monoids" rel="nofollow">Hom alg for comm. monoids</a>. See also the related questions: <a href="http://mathoverflow.net/questions/14266/a-peculiar-model-structure-on-simplicial-sets" rel="nofollow">A peculiar model strcture on simplicial sets?</a> and <a href="http://mathoverflow.net/questions/13518/is-in-simplicial-commutative-monoids-group-completion" rel="nofollow">simplicial commutative monoids group completion</a>. The example I just described also works to give a simplicial commutative monoid where now X is any simplicial set. However when you apply the "Dold-Kan correspondence" you always get the zero chain complex. This shows that the Dold-Kan correspondence fails to be an equivalence for commutative monoids. It also says that in order to describe higher categories in terms of something like a chain complex (e.g. something like a crossed module) you absolutely need some invertablity. </p>