Can one drop the retracts in the definition of anodyne extensions?

Question

Definition: Anodyne extensions(i.e. acyclic cofibration) of simplicial sets are the closure of horn inclusions under transfinite composition, pushouts, and retracts.

The composition and pushouts can be easily thought of as attaching cells. My question: can we drop the requirement of retracts? Do we have a conterexample show that retract is must needed?

Answer 1

There are anodyne extensions that are not obtained using only pushouts and coproducts. The following example is due to my colleague.

Let $p$ be the composite $\Lambda^2_0 \to \Delta^2 \overset{s^1}{\to} \Delta^1$, where $s^1 : \Delta^2 \to \Delta^1$ is the morphism corresponding to the (degenerate) 2-simplex $0 \to 1 \to 1$. Form the following pushout square: $$\require{AMScd} \begin{CD} \Lambda^2_0 @>{p}>> \Delta^1 \\ @VVV @VVV \\ \Delta^2 @>>{x}> X \end{CD}$$ Let $d^0 : \Delta^1 \to \Delta^2$ be the morphism corresponding to the edge $1 \to 2$, let $e : \Lambda^2_1 \to X$ be the composite $\Lambda^2_1 \to \Delta^2 \overset{x}{\to} X$ and form the following pushout square: $$\begin{CD} \Lambda^2_1 @>{e}>> X \\ @VVV @VVV \\ \Delta^2 @>>{y}> Y \end{CD}$$ By construction, $\Delta^1 \to X \to Y$ is an anodyne extension, so by the 2-out-of-3 property, $Y \to \Delta^0$ is a weak homotopy equivalence, and hence any morphism $\Delta^1 \to Y$ whatsoever is a weak homotopy equivalence.

Let $d^1 : \Delta^1 \to \Delta^2$ be the morphism corresponding to the edge $0 \to 2$. Then composite $\Delta^1 \overset{d^1}{\to} \Delta^2 \overset{y}{\to} Y$ is an anodyne extension, and it cannot be obtained using only coproducts and pushouts of horn inclusions. To see this, it is useful to have a more explicit description of $Y$:

• $Y$ has two vertices, say $a$ and $b$.
• $Y$ has three non-degenerate edges: $\alpha, \gamma : a \to b$ and one edge $\beta : b \to b$.
• $Y$ has two non-degenerate 2-simplices: $x$, with edges $\beta, \alpha, \alpha$, and $y$, with edges $\beta, \gamma, \alpha$.

The anodyne extension in question is the morphism $f : \Delta^1 \to Y$ corresponding to $\gamma$. If it were possible to express $f$ using coproducts and pushouts of horn inclusions, then we must only use horn inclusions of the form $\Lambda^2_k \to \Delta^2$; but one can verify that there is no possible commutative diagram of the form below, $$\begin{CD} \Lambda^2_k @>>> \Delta^1 @>{f}>> Y \\ @VVV @VVV @| \\ \Delta^2 @>>> Y' @>>> Y \end{CD}$$ where the left square is a pushout square and $Y' \to Y$ is a monomorphism. This proves the claim.