Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:00:11Z http://mathoverflow.net/feeds/question/49956 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49956/historical-and-terminological-questions-about-dan-kans-ex-functor-and-its-relati Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes Harry Gindi 2010-12-20T12:22:50Z 2010-12-20T17:14:08Z <p>Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as well, but this follows from injectivity) partially ordered by inclusion. Composing this with the obvious inclusion $\iota:\operatorname{Poset}\hookrightarrow \operatorname{Cat}$ and the nerve $\operatorname{Cat}\to \operatorname{sSet}$, this yields a functor $\Delta\to \operatorname{sSet}$, and by the universal property of presheaf categories, there exists a unique colimit-preserving lift $\operatorname{Sd}:\operatorname{sSet}\to \operatorname{sSet}$ By adjoint functor nonsense, we obtain an adjunction $$\operatorname{Sd}:\operatorname{sSet}\leftrightarrows \operatorname{sSet}:\operatorname{Ex}.$$ </p> <p>The functor $\operatorname{Sd}$, unsurprisingly is called the <em>barycentric subdivision</em>, since we obtain an isomorphism $$\operatorname{Sd}\circ N_{\mathcal{CS}}(-)\cong N_{\mathcal{CS}}\circ \xi_\mathcal{CS}(-)$$ of functors $\mathcal{CS}\to sSet$ where $\mathcal{CS}$ is the classical category of combinatorial simplicial complexes, $N_\mathcal{CS}:\mathcal{CS}\to sSet$ is the nerve of simplicial complexes${}^1$, and $\xi_{\mathcal{CS}}:\mathcal{CS}\to \mathcal{CS}$ is the classical barycentric subdivision of simplicial complexes. </p> <p>Where does the two-letter abbreviation for $\operatorname{Ex}$ come from? Does it actually stand for an english word? Does it have a classical analogue for combinatorial <em>or</em> topological simplicial complexes in the way that barycentric subdivision does?</p> <p>(I wasn't sure if I should tag this as a soft-question, so I added some gratuitous background and a mathematical question to even things out. I have marked this question community wiki, however, since I would probably call for wikification if the question were asked by somebody else).</p> <hr> <p>Notes:<br> $({}^1)$ To do this, fix a strict total ordering $\tau$ of the vertices $E$ of the simplicial complex $X=(E,X_\Delta)$. Consider $X^\tau$ as an object of the slightly more strict category of <em>oriented simplicial complexes</em>, denoted by $\mathcal{OCS}$, which consists of simplicial complexes with strict total orders on their sets of vertices and orientation-preserving (that is to say, order-preserving on vertices) morphisms between them. Then we define the <em>oriented realization</em> <code>$\Delta^\bullet_\mathcal{OCS}:\Delta\to \mathcal{OCS}$</code> by the formula $[\bullet]\mapsto \Delta^\bullet_\mathcal{OCS}:=([\bullet],\xi([\bullet]))$. Then define </p> <p><code>$$N_\mathcal{OCS}(X^\tau)_n:=Hom_{\mathcal{OCS}}(\Delta^n_\mathcal{OCS},X^\tau).$$</code></p> <p>If we're willing to include orientation in our definition of a simplicial complex, then we may drop the $\mathcal{O}$, and the isomorphism of functors that we noted actually exists. If not, then the statement only holds up to choosing orderings (in particular, $N_\mathcal{CS}$ depends heavily on the orientation, but picking an orientation is "not functorial on morphisms" (whatever that may mean!)). </p>