Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes

2010-12-20T12:22:50Z

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 $$\operatorname{Sd}:\operatorname{sSet}\leftrightarrows \operatorname{sSet}:\operatorname{Ex}.$$

The functor $\operatorname{Sd}$, unsurprisingly is called the barycentric subdivision, 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.

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 or topological simplicial complexes in the way that barycentric subdivision does?

(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).

Notes:
$({}^1)$ 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 oriented simplicial complexes, 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 oriented realization $\Delta^\bullet_\mathcal{OCS}:\Delta\to \mathcal{OCS}$ by the formula $[\bullet]\mapsto \Delta^\bullet_\mathcal{OCS}:=([\bullet],\xi([\bullet]))$. Then define

$$N_\mathcal{OCS}(X^\tau)_n:=Hom_{\mathcal{OCS}}(\Delta^n_\mathcal{OCS},X^\tau).$$

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!)).

Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes - MathOverflow

Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes