nLab crossed group

Idea

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

Definition

Definition

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

1. for each object $s \in S$ a group structure on $G_s$;

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

such that for all morphisms $\alpha : s \to r$ and $\beta : t \to s$ in $S$ and $g,h \in G_r$ we have

1. $g_*( \alpha \circ \beta) = g_*(\alpha) \circ \alpha^*(g)_* \beta$;

2. $g_* (id_r) = id_r$;

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

4. $\alpha^*(e_r) = e_s$;

where $g_*$, $h_*$ denotes the group action and

$\alpha^* : G_r \to G_s$ the presheaf map.

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

$$(\alpha, g) \circ (\beta, h) = (\alpha \cdot g_*(\beta), \beta^*(g) \cdot h) \,.$$

(Ber-Moe, def. 2.1).

Examples

• In case, $G$ is constant, i.e. $G_s=G$ and $\alpha^*(g)=g$ for all $s,\alpha \in S$, $S G$ reduces to the Grothendieck construction for the functor $G\to Cat$ that picks out $S$.

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

Properties

Definition

If $S$ is equipped with a generalized Reedy structure, then an $S$-crossed group $G$ is called compatible with that generalized Reedy structure if

1. the $G$-action respects $S^+$ and $S^-$;

2. if $\alpha : r \to s$ is in $S^-$ and $g \in G_s$ such that $\alpha^* (g) = e_r$ and $g_*(\alpha) = \alpha$, then $g = e_s$.

Proposition

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

(Ber-Moe, prop. 2.10).

Remarks

• A related notion is “thickening” (cf. Cisinski (8.5.8; 2006), Isaacson (2010)).

Related entries

• skew-simplicial set

• generalized Reedy category

• crossed homomorphism

• cyclic set

• symmetric set

• quaternionic set

• dihedral set

References

• Michael Batanin, Martin Markl, Crossed interval groups and operations on the Hochschild cohomology, arXiv:0803.2249 (2008). (abstract)

• Clemens Berger, Ieke Moerdijk, On an extension of the notion of Reedy category (2008) (arXiv:0809.3341)

• Denis-Charles Cisinski, Les préfaisceaux comme modèles des types d'homotopie, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages (pdf)

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

• B. L. Feigin, B. L. Tsygan, Additive K-theory, pp.67-203 in LNM 1289 Springer Heidelberg 1986.

• Zbigniew Fiedorowicz, Jean-Louis Loday, Crossed simplicial groups and their associated homology, Trans. AMS 326 (1991) 57-87 [doi:10.1090/S0002-9947-1991-0998125-4]

• Samuel Isaacson, Symmetric cubical sets, Journal of Pure and Applied Algebra 215 (2011). (arXiv, doi:10.1016/j.jpaa.2010.08.001)

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

• Jean-Louis Loday, Cyclic Homology, Springer Heidelberg 1998$^2$.

• Walker H. Stern, Structured topological field theories via crossed simplicial groups, arXiv:1603.02614 (2016). (abstract)

• Jun Yoshida, A general method to construct cube-like categories and applications to homotopy theory, arXiv:1502.07539. (abstract)

• Jun Yoshida, Colimits and limits of crossed groups, arXiv:1802.06644 (2018). (abstract)