The category Δ\Delta has generators (cofaces and codegeneracies) and relations and a every morphism can be canonically represented via cofaces and codegeneracies. Consequently, for simplicial sets, hence functors Δ opSet\Delta^{op}\to Set have canonical representation via face and degeneracy maps. We can now look at categorification of this situation to pseudofunctors, say Δ opCat\Delta^{op}\to Cat (pseudosimplicial categories). While the simplicial identities now hold up to invertible 2-cells, the canonical representation of general morphisms as sequences of face and degeneracy maps suggest that one should be able to write down the corresponding 2-cells from generating 2-cells corresponding to the quadratic identities, and that different choices of sequences generating 2-cells should be the same composite 2-cells.

Jardine has proved that the generating 2-cells

j i i j+1 (ij) iσ jσ j1 i (i<j) iσ jId (i=j,i=j+1) iσ jσ j i1 (i>j+1) σ iσ jσ j+1σ i (ij)\array{ \partial_j \partial_i \implies \partial_i \partial_{j+1} & (i \leq j)\\ \partial_i \sigma_j \implies \sigma_{j-1} \partial_i & (i \lt j) \\ \partial_i \sigma_j \implies Id & (i = j, i = j + 1)\\ \partial_i \sigma_j \implies \sigma_j \partial_{i-1} & (i \gt j + 1) \\ \sigma_i \sigma_j \implies \sigma_{j+1} \sigma_i & (i \leq j) }

coming from the pseudosimplicial objects satisfy 17 coherence relations which in turn enable one to reconstruct the 2-cells among more complicated chains of face and degeneracy maps in unique way. To write those, all generating will be denoted by α\alpha with some additional superscripts, but for simplicity we skip the superscript as in Jardine’s paper (as they are obvious to fill):

In particular, this way one can check when one has a pseudosimplicial object coming from a concrete construction by producing just those 2-cells which are generating and checking the 17 diagrams. He calls the corresponding structure (X n, i,σ j,α)(X_n, \partial_i, \sigma_j, \alpha) of 0-cells, faces, degeneracies and generating 2-cells the supercoherence. They are therefore equivalent to a pseudosimplicial object X X_\bullet.

  • J. F. Jardine, Supercoherence, J. Pure Appl. Algebra 75 (1991), no. 2, 103–194 MR1138365 doi

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