nLab thin element

Idea

“Thin elements” occur in various contexts. The basic idea is that many higher categorical structures have cells (of various shapes). However, in some contexts, some of these cells should be regarded as “more canonical,” “more equivalence-like,” or “more identity-like” than others. They are ‘’really’‘ there because of lower dimensional features. The cells with this property are frequently called thin and the extra information of which cells are thin is called a thin structure or a stratification.

Thin cells in higher groupoids

In the simplicial singular complex $S X$ of a topological space $X$, a singular simplex $f: \Delta^n \to X$ is called thin if it factors through a retraction $r: \Delta^n \to \Lambda^{n-1}_i$ to some horn of $\Delta^n$. Then the well known Kan condition on $S X$ can be strengthend to say that every horn in $S X$ has a thin filler. There are also, usually, other non-thin fillers which depend on $X$.

It is difficult to decide the laws that these these thin elements should satisfy. Keith Dakin’s axioms on thin elements in a simplicial set $K$ with subsets $T_n \subseteq K_n, n \geq 1$, of elements called thin are

1. Degenerate elements are thin;

2. Any horn has a unique thin filler;

3. If all faces but one of a thin element are thin, then so is the remaining face.

Ashley’s theorem, conjectured by Dakin, is that the category of such $T$-complexes is equivalent to that of crossed complexes.

As for relations with complexes having cells of different shapes, Brown and Higgins proved that cubical $T$-complexes are equivalent to crossed complexes, and to strict cubical $omega$-groupoids with connections. In that theory, the notion of thin element is crucial because it can be proved that any (multiple) composition of thin elements is thin, and because of the relation of thin elements to commutative shells. David Jones in his thesis developed the notion of polyhedral $T$-complex, and gave sufficient axioms for these to be equivalent to crossed complexes.

Thin structures in double categories

Among other things, a thin structure on a double category can be seen as giving the information of which squares should be thought of as commuting rather than simply containing a 2-cell. The most general definition seems to be that a thin structure on a double category $D$ is a morphism of double categories $\Theta: \square C \to D$ where $C$ is any category and $\square C$ is the double category of commuting squares in $C$.

Note that $D$ contains the categories $D_v, D_h$ of vertical and horizontal edges respectively, and it is often convenient to take $C$ to be say $D_v$ and for $\Theta$ to be the identity on $D_v$. Then $\Theta$ defines a function $\Gamma: D_v \to D$ to the set of squares of $D$ which is called a connection, and also a functor $\gamma: D_v \to D_h$ which is called the holonomy of the connection $\Gamma$. The laws which $\Gamma$ satisfies allow $\Theta$ to be recovered from $\Gamma$ in some cases.

Thus some thin structures contain a way to represent any vertical arrow as a horizontal arrow (or vice versa, depending on one’s conventions). Thin structures can equivalently be described by “foldings” and “connection pairs.” It turns out that when they exist, they are unique up to a (suitably weak) notion of equivalence; see also framed bicategory.

Simplicial nerves for higher categories

Following on the work mentioned above for thin elements in Kan complexes, one can consider thin elements in nerves of higher categories. In “The algebra of oriented simplices” Street constructed the simplicial nerve for a strict omega-category, which comes equipped with a collection of thin simplices that are identities (rather than merely cells). Thus this nerve is not just a simplicial set, but a stratified simplicial set (a simplicial set equipped with a collection of “thin simplices” containing all degeneracies).

Street and Roberts conjectured that the nerves of strict $\omega$-categories could be characterized as stratified simplicial sets such that 1. every complicial horn? has a unique thin filler, and 1. for such a thin filler, if all faces but the one missing face in the complicial horn are thin, then so is the last one.

Such stratified simplicial sets are called complicial sets. This conjecture was later proved by Verity.

A weak complicial set is a stratified simplicial set satisfying the above conditions except that thin fillers need not be unique. Street conjectured that these represent the nerves of weak $\omega$-categories, where now the thin simplices are supposed to be the equivalences rather than the identities. But since every complicial set is weak complicial, one should also require that the collection of thin simplices is “maximal” in some sense; Street gave one proposed characterization. Weak complicial sets have been studied extensively by Verity.

References

Relevant references for thinness in the context of simplicial T-complexes include:

• Ronnie Brown, An Introduction to simplicial T-complexes, Esquisses Math. 32 (1983) Part 1

• M.K. Dakin, Kan complexes and multiple groupoid structures Ph.D Thesis, University of Wales, Bangor, 1977. Esquisses Math. (1983) 32 Part 2

• N. Ashley, Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan, Ph.D Thesis University of Wales, Bangor, 1978; Dissertationes Math., 165, (1989), 11 – 58. Esquisses Math. (1983) 32 Part 3

Verity’s work can be found in

• D. R. Verity, 2005, Complicial Sets , available from : arXiv:math.CT/0410412.

• D. Verity, 2006, Weak complicial sets I: basic homotopy theory , available from : arXiv:math/0604414.

• D. R. Verity, 2006, Weak complicial sets. III. Enriched and internal quasi-category theory , (in preparation).

• D. R. Verity, 2007, Weak complicial sets. II. Nerves of complicial Gray-categories , in Categories in algebra, geometry and mathematical physics , volume 431 of Contemp. Math., 441–467, Amer. Math. Soc., Providence, RI.