nLab quaternionic set

Idea

A quaternionic set is a simplicial set with a quarternionic symmetry, in other words it is a presheaf on the quaternionic simplex category $\Delta Q$ which contains the ordinary simplex category $\Delta$ and an opposite quaternion group $Q_{n+1}^{op}$ as the automorphism group at the objects $[n]$, whence quaternionic sets are an example of an crossed simplicial object.

Definition

Let $Q_n$ be the group given by the presentation $\{x,y| x^n = y^2\;,y x y^{-1}=x^{-1}\}$, hence the $n$th quaternion group, also called a binary dihedral group.

Definition

The quaternionic simplex category¹ $\Delta Q$ has objects $[n],\; n\geq 0\;$, contains the (topologists’) simplex category $\Delta$ as a (non-full) wide subcategory and each object $[n]$ has as automorphism group the (opposite of the) $n+1$st quaternion group: $Aut_{\Delta Q}([n]) = Q_{n+1}^{op}$.

Furthermore, any morphism $\varphi:[m]\to [n]$ in $\Delta Q$ can be uniquely factored as $\varphi = s\circ q$ with $s:[m]\to [n]\in \Delta$ and $q\in Q_{m+1}^{op}$.

A quaternionic set is a presheaf on $\Delta Q$.

Properties

• According to the classification of crossed simplicial groups, $\Delta Q$ or, more precisely, the equivalent crossed group $Q_*$ corresponds to the exact sequence of crossed groups $1\to \mathbb{Z}/2\to Q_*\to D_*\to 1\; .$ Whence it is of dihedral type and, accordingly, the model category techniques for planar crossed simplicial groups apply (cf. Spaliński, Balchin).

• The geometric realization of $Q_*$ is the normalizer of $S^1$ in $S^3$ (cf. Loday).

• Just like in the cyclic and dihedral cases, $\Delta Q$ is self-dual: $\Delta Q\cong \Delta Q^{op}$ (cf. Dunn, prop.1.4).

• As usual for crossed groups (or, more generally, for functors between small categories that are surjective on objects; cf. Moerdijk-Maclane, pp.359, 377f), the inclusion $i:\Delta\to \Delta Q$ induces by Kan extension an essential surjective geometric morphism $i_!\vdash i^*\vdash i_*: Set^{\Delta^{op}}\to Set^{\Delta Q^{op}}\;$. Here the inverse image $i^*:Set^{\Delta Q^{op}}\to Set^{\Delta^{op}}$ restricts a presheaf $P$ on $\Delta Q$ to one on $\Delta$ by pre-composition: $i^*(P)=P\circ i^{op}\,$.

• Since $i:Set^{\Delta^{op}}\to Set^{\Delta Q^{op}}$ is surjective, it follows again by generalities that $Set^{\Delta Q^{op}}$ is (equivalent to) the category of algebras for the monad induced by $i^*i_!$ on $Set^{\Delta^{op}}$ whence quaternionic sets are indeed simplicial sets enhanced with a quaternionic algebra structure. From this perspective, $i_!\vdash i^*$ is an adjunction between a free algebra functor and an underlying (simplicial) set functor. (For an explicit description of the free skew-simplicial set on a simplicial set and the corresponding monad for general crossed groups $G_*$ see Krasauskas.)

• $\Delta Q$ is an Eilenberg-Zilber category with degree function $deg:\Delta Q\to \mathbb{N}$ inherited from $\Delta$. The same holds for other types of skew simplicial sets, or more generally, for crossed groups over a strict Reedy category (cf. Berger-Moerdijk ex. 6.8).

• Truncating $\Delta Q$ at $[1]$ and taking presheaves, one obtains a category of quaternionic graphs $Set^{\Delta Q_{\leq 1}^{op}}$ with a $Q_2$ action, just like reflexive graphs result from $\Delta_1$, resp. reversible graphs from truncating $\Delta S$ in the symmetric simplex category.

Related entries

• skew-simplicial set

• crossed group

• cyclic set

• dihedral set

• quaternion

References

• Jean-Louis Loday, Homologies diédrale et quaternionique, Adv. in Math. 66 (1987) pp.119-148.

• Gerald Dunn, Dihedral and Quaternionic Homology and Mapping Spaces, K-theory 3 (1989) pp.141-161.

• Jan Spaliński, Homotopy theory of dihedral and quaternionic sets, Topology 39 (1995) pp.35-52.

• R. Krasauskas, Skew-simplicial groups, Lithuanian Math. J. 27 pp.47-54 (1987).

• Zbigniew Fiedorowicz, Jean-Louis Loday, Crossed simplicial groups and their associated homology, Trans. AMS 326 pp.57-87 (1991).

• Clemens Berger, Ieke Moerdijk, On an extension of the notion of Reedy category, Math. Z. 269 (2011) pp.977-1004. (arXiv:0809.3341)

• Tobias Dyckerhoff, Mikhail Kapranov, Crossed simplicial groups and structured surfaces, arXiv:1403.5799 (2014). (abstract)

• Scott Balchin, Homotopy of planar Lie group equivariant presheaves, J. Homotopy Relat. Struct. 13 (2018) pp.555-580. (doi)

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1. Since $\Delta Q$ encapsulates exactly the structure of $Q_*$ (i.e. the simplicial set that has fibers $Q_{n+1}$ at $[n]$ together with simplicially compatible actions) it is also referred to as the quaternionic crossed simplicial group. The terminology used at crossed group would refer to it as the total category of the crossed simplicial group $Q_*$. Note also that we follow Loday by taking $Q_n^{op}$ as automorphism group at $[n-1]$ whereas other authors like Krasauskas or Dyckerhoff-Kapranov use $Q_n$. ↩