Cofibrant Simplicial Sets

1. Elegant Reedy categories 1.1. Reedy categories. Recall that a Reedy structure on a category C is given by: a pair of identity-on-objects subcategories C− and C+ of C together with a degree function |–| : ob C → N such that: • For all non-identity σ : c→ d in C−, we have |c| > |d|; • For all non-identity δ : c→ d in C+, we have |c| < |d|. • Every map γ : c→ d of C admits a unique factorisation γ = δσ where σ ∈ C− and δ ∈ C+. 1.2. Presheaves and degenerate elements. Given a presheaf X on a Reedy category C, we will write xγ for the action of a map γ : d→ c of C on an element x ∈ X(c); thus xγ = (Xγ)(x) ∈ X(d). We say that x ∈ X(c) is non-degenerate if, whenever x = yσ with σ ∈ C−, we have σ = 1c, and say that it is degenerate if x = yσ for some non-identity σ : d→ c in C−. We write Xnd(c) and Xd(c) for the sets of non-degenerate and degenerate elements of X(c). 1.3. Elegant Reedy categories. A Reedy category C is called elegant ([]) if, for every presheaf X : Cop → Set and x ∈ X(c), there is a unique pair (σx ∈ C−(d, c), x̄ ∈ Xnd(d)) with x = x̄σx. Proposition 1. C is an elegant Reedy category if and only if every span of maps in C− can be completed to a commutative square in C− which is an absolute pushout in C. It follows that every map in C− is a split epimorphism. Proof. See Proposition 3.8 of [] and the remarks following. 2. Categories of non-degenerate elements 2.1. Degeneracy-reflecting maps. It is immediate that a map of presheaves f : X → Y on a Reedy category C preserves degeneracy and reflects non-degeneracy, in the sense that f(Xd(c)) ⊆ Yd(c) and f(Ynd(c)) ⊆ Xnd(c) for all c ∈ C. We say that a map of presheaves f : X → Y on a Reedy category reflects degeneracy or preserves non-degeneracy if one of the two equivalent conditions f(Xnd(c)) ⊆ Ynd(c) and f(Yd(c)) ⊆ Xd(c) Date: April 5, 2013. 2000 Mathematics Subject Classification. Primary: 18D05, 18C15. The author acknowledges the support of an Australian Research Council Discovery Project, grant number DP110102360. 1