categorical wreath product

for disambiguation see wreath product




Let AA be a small category. Its categorical wreath product with the simplex category is the category ΔA\Delta \wr A whose

  • objects are kk-tuples ([k],(a 1,,a k))([k], (a_1, \cdots, a_k)) of objects of AA, for any kk \in \mathbb{N};

  • morphisms are tuples

    (ϕ,ϕ ij):([k],(a 1,,a k))([l],(b 1,,b l)) (\phi, \phi_{i j}) : ([k],(a_1, \cdots, a_k)) \to ([l],(b_1, \cdots, b_l))

    consisting of

    • a morphism ϕ:[k][l]\phi: [k] \to [l] in Δ\Delta;

    • morphisms ϕ ij:a ib j\phi_{i j} : a_i \to b_j for 0<ik0 \lt i \leq k and ϕ(i1)<jϕ(i)\phi(i-1) \lt j \leq \phi(i).

(Berger, def. 3.1).


An object of ΔA\Delta \wr A is to be thought of as a sequence of morphisms labeled by objects of AA

0 a 1 1 a 2 a n n \array{ 0 \\ \downarrow \mathrlap{a_1} \\ 1 \\ \downarrow \mathrlap{a_2} \\ \downarrow \\ \vdots \\ \downarrow \mathrlap{a_n} \\ n }

and morphisms are given by maps between these linear orders equipped with morphisms from the kkth object in the source to all the objects in the target that sit in between the image of the kkth step.



Section 3 of