By M. Anderson

Summary algebra textbook

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X) = j ∗ τd (i! (X)) for every simplicial set X. In particular, there is a diagram of adjoint functors sSets O o τ / i! τd N Cat o dSets O i∗ / j! Nd Oper j∗ with left adjoints on top or to the left. Moreover, the following commutativity relations hold up to natural isomorphisms: τ N = id, i∗ i! = id, τd Nd = id, j ∗ j! = id, and j! τ = τd i! , N j ∗ = i∗ Nd , i! N = Nd j! There is also a column in the middle of the square relating planar dendroidal sets with non-Σ operads. 5. Not everything commutes in the above diagram.

Using the adjunction one can check that also in ∗ in∗ (X) = X for all X. It follows that in! and in∗ are both fully faithful. 2. A dendroidal set X is called n-coskeletal if X = coSkn (X). 3. A dendroidal set X is n-coskeletal if, for every dendroidal set Y , each map Skn (Y ) −→ X extends uniquely along Skn (Y ) −→ Y to a map Y −→ X. Proof. Since X is n-coskeletal, X = coSkn (X). By an adjointness argument, there is a bijection between the sets of maps dSets(Y, X) and dSets(Skn (Y ), X). If we make the definition of the Kan extension in!

In other words, Skn (X)R consists of equivalence classes of pairs (α, u), with u : R −→ T in Ω and α : Ω[T ] −→ X in dSets, and |V (T )| ≤ n. The equivalence relation on such pairs is generated by (αv, u) ∼ (α, vu) u v α where R −→ T −→ T and Ω[T ] −→ X. The counit maps the equivalence class of (α, u) to u∗ (α), or, in another notation, it maps (α, u) to the composition α ◦ u, u α Ω[R] −→ Ω[T ] −→ X. 4. For each n ≥ 0, the counit of the adjunction Skn (X) −→ X is a monomorphism for every dendroidal set X.