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Consider the map $$i\colon A\sqcup A\to A$$ where $i=(Id_{A},Id_{A})$. Is this map a cofibration? Actually, I know that isomorphisms are cofibrations and pushouts of cofibrations are cofibrations. But I do not know what tools to use to prove that this map is a cofibration (if it is).

T. Wildwolf
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No, this is false. Consider the standard model structure on simplicial sets. The cofibrations are precisely the monomorphisms in this structure. However, unless $A$ is the empty simplicial set, your $i$ is never a monomorphism.

FShrike
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  • Oh thanks. My question arises because I don't know what cylinder object I should take for two equal morphisms $f=g\colon A\to B$ (they should be left homotopic if they are equal). – T. Wildwolf Aug 02 '23 at 16:16
  • @T.Wildwolf In any model category, at least one 'very good' cylinder object $\mathrm{Cyl}(A)$. Write $p:\mathrm{Cyl}(A)\to A$ for the trivial fibration and $i_0,i_1:A\to\mathrm{Cyl}(A)$ for the weak equivalences. If $f=g$, then we have a very good left homotopy $h:\mathrm{Cyl}(A)\to B$ given by $f\circ p=g\circ p$, witnessing $f\simeq_L g$ – FShrike Aug 02 '23 at 16:24
  • (If $A$ is cofibrant, then (good) left homotopy is an equivalence relation) – FShrike Aug 02 '23 at 16:27
  • Could you mention a reference for this good cylinder object? I cannot understand how to construct it, I want to study the full details. – T. Wildwolf Aug 02 '23 at 21:43
  • @T.Wildwolf A "very good" cylinder object for $A$ is an object $\mathrm{Cyl}(A)$ equipped with a trivial fibration $p:\mathrm{Cyl}(A)\to A$ and weak equivalences $i_0,i_1:A\to\mathrm{Cyl}(A)$ which make $i_0\sqcup i_1:A\sqcup A\to\mathrm{Cyl}(A)$ a cofibration and satisfy $pi_0=1_A=pi_1$. To obtain this, all you do is factor the codiagonal $\nabla:A\sqcup A\to A$ (in your notation, this is "$i$") as $A\sqcup A\overset{j}{\longrightarrow}\mathrm{Cyl}(A)\overset{p}{\longrightarrow}A$ where $p$ is a trivial fibration and $j$ is a cofibration. – FShrike Aug 02 '23 at 22:03
  • You define $i_0,i_1$ as the two composites $A\hookrightarrow A\sqcup A\overset{j}{\longrightarrow}\mathrm{Cyl}(A)$. – FShrike Aug 02 '23 at 22:03
  • Such a factorisation exists by the axioms on model categories – FShrike Aug 02 '23 at 22:03
  • So, for any object $A$, at least one "very good" cylinder exists. Similarly at least one "very good" cocylinder exists. – FShrike Aug 02 '23 at 22:04
  • ("very good cylinder" is distinct from "good cylinder" is distinct from "cylinder", similarly "very good left homotopy" is distinct from "good left homotopy" is distinct from "left homotopy". That said, if $f$ is left homotopic to $g$ there must also be a good left homotopy from $f$ to $g$) – FShrike Aug 02 '23 at 22:07