nLab crossed group



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Basic facts


(,1)(\infty,1)-Category theory



A crossed group is a presheaf with additional symmetry given by a fiberwise group structure and actions on the morphisms of the underlying category.



For SS a small category, a crossed SS-group is a presheaf G:S opSetG : S^{op} \to Set equipped with

  1. for each object sSs \in S a group structure on G sG_s;

  2. for all s,rSs, r\in S a left G rG_r-action on the hom-set S(s,r)S(s,r) ;

such that for all morphisms α:sr\alpha : s \to r and β:ts\beta : t \to s in SS and g,hG rg,h \in G_r we have

  1. g *(αβ)=g *(α)α *(g) *βg_*( \alpha \circ \beta) = g_*(\alpha) \circ \alpha^*(g)_* \beta;

  2. g *(id r)=id rg_* (id_r) = id_r;

  3. α *(gh)=h *(α) *(g)α *(h)\alpha^* (g \cdot h) = h_*(\alpha)^*(g)\cdot \alpha^*(h);

  4. α *(e r)=e s\alpha^*(e_r) = e_s;

where g *g_*, h *h_* denotes the group action and

α *:G rG s\alpha^* : G_r \to G_s the presheaf map.

The total category SGS G of an crossed SS-group GG is the category with the same objects as SS, and with morphisms rsr \to s being pairs (α,g)S(s,r)×G r(\alpha, g) \in S(s,r)\times G_r and with composition defined by

(α,g)(β,h)=(αg *(β),β *(g)h). (\alpha, g) \circ (\beta, h) = (\alpha \cdot g_*(\beta), \beta^*(g) \cdot h) \,.

(Ber-Moe, def. 2.1).


  • In case, GG is constant, i.e. G s=GG_s=G and α *(g)=g\alpha^*(g)=g for all s,αSs,\alpha \in S, SGS G reduces to the Grothendieck construction for the functor GCatG\to Cat that picks out SS.

  • Let Ω pl\Omega_{pl} be the category of finite planar trees. Then the category Ω\Omega of non-planar finite rooted trees arises as the total category of an Ω pl\Omega_{pl}-crossed group which to a planar tree TT assigns its group of non-planar automorphisms. This example is somewhat typical, since Ω pl\Omega_{pl} is a strict Reedy category where all automorphisms are trivial which gets enhanced with non-trivial symmetries to Ω\Omega, now a generalized Reedy category (cf. below).



If SS is equipped with a generalized Reedy structure, then an SS-crossed group GG is called compatible with that generalized Reedy structure if

  1. the GG-action respects S +S^+ and S S^-;

  2. if α:rs\alpha : r \to s is in S S^- and gG sg \in G_s such that α *(g)=e r\alpha^* (g) = e_r and g *(α)=αg_*(\alpha) = \alpha, then g=e sg = e_s.


Let SS be a strict Reedy category and let GG be a compatible SS-crossed group. Then there exists a unique dualizabe generalized Reedy structure on SGS G for which the embedding SSGS \hookrightarrow S G is a morphism of generalized Reedy categories.

(Ber-Moe, prop. 2.10).



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