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

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!)).

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I do not understand your composition \Delta \to sSet unless your nerve functor should be Cat \to sSet. –  Sean Tilson Dec 20 '10 at 16:12
${\rm Ex}$ for extension (see Kan, "On c.s.s. complexes", Am. J. Math 79, vol. 3 (1957), pp 449--476. In particular section 3). –  Theo Buehler Dec 20 '10 at 16:54
@Sean: Yes, that was a typo. (I copied and pasted "\operatorname{sSet}" and forgot to swap out the middle on one of them! –  Harry Gindi Dec 20 '10 at 17:14
For the terminology question, let me add: what most people call a "Kan complex", Kan calls in this paper a simplicial set satisfying the "extension condition". $Ex^\infty(X)$ is a replacement of $X$ which satisfies the EXtension condition. –  Charles Rezk Dec 20 '10 at 18:18