nLab quaternionic set

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Topos theory

topos theory

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In higher category theory

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Contents

Idea

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

Definition

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

Definition

The quaternionic simplex category1 ΔQ\Delta Q has objects [n],n0[n],\; n\geq 0\;, contains the (topologists’) simplex category Δ\Delta as a (non-full) wide subcategory and each object [n][n] has as automorphism group the (opposite of the) n+1n+1st quaternion group: Aut ΔQ([n])=Q n+1 opAut_{\Delta Q}([n]) = Q_{n+1}^{op}.

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

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

Properties

  • According to the classification of crossed simplicial groups, ΔQ\Delta Q or, more precisely, the equivalent crossed group Q *Q_* corresponds to the exact sequence of crossed groups 1/2Q *D *1.1\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 *Q_* is the normalizer of S 1S^1 in S 3S^3 (cf. Loday).

  • Just like in the cyclic and dihedral cases, ΔQ\Delta Q is self-dual: ΔQΔQ op\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:ΔΔQi:\Delta\to \Delta Q induces by Kan extension an essential surjective geometric morphism i !i *i *:Set Δ opSet ΔQ opi_!\vdash i^*\vdash i_*: Set^{\Delta^{op}}\to Set^{\Delta Q^{op}}\;. Here the inverse image i *:Set ΔQ opSet Δ opi^*:Set^{\Delta Q^{op}}\to Set^{\Delta^{op}} restricts a presheaf PP on ΔQ\Delta Q to one on Δ\Delta by pre-composition: i *(P)=Pi opi^*(P)=P\circ i^{op}\,.

  • Since i:Set Δ opSet ΔQ opi:Set^{\Delta^{op}}\to Set^{\Delta Q^{op}} is surjective, it follows again by generalities that Set ΔQ opSet^{\Delta Q^{op}} is (equivalent to) the category of algebras for the monad induced by i *i !i^*i_! on Set Δ opSet^{\Delta^{op}} whence quaternionic sets are indeed simplicial sets enhanced with a quaternionic algebra structure. From this perspective, i !i *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 *G_* see Krasauskas.)

  • ΔQ\Delta Q is an Eilenberg-Zilber category with degree function deg:ΔQdeg:\Delta Q\to \mathbb{N} inherited from Δ\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 ΔQ\Delta Q at [1][1] and taking presheaves, one obtains a category of quaternionic graphs Set ΔQ 1 opSet^{\Delta Q_{\leq 1}^{op}} with a Q 2Q_2 action, just like reflexive graphs result from Δ 1\Delta_1, resp. reversible graphs from truncating ΔS\Delta S in the symmetric simplex category.

References


  1. Since ΔQ\Delta Q encapsulates exactly the structure of Q *Q_* (i.e. the simplicial set that has fibers Q n+1Q_{n+1} at [n][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 *Q_*. Note also that we follow Loday by taking Q n opQ_n^{op} as automorphism group at [n1][n-1] whereas other authors like Krasauskas or Dyckerhoff-Kapranov use Q nQ_n.

Last revised on April 28, 2024 at 14:41:44. See the history of this page for a list of all contributions to it.