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In the excellent "A Handbook of Model Categories" (2021), cofibrant and fibrant homotopies are defined exactly as it seemed natural to me: immediately through functorial cylindrical objects / path space objects (hereinafter I will only talk about cylindrical objects), and the concept of non-functorial cylindrical objects in general not mentioned. But for proofs of elementary statements such as "cofibrate homotopy is an equivalence relation if dom is a cofibrant" Scott Balchin refers to Hovey, who proves symmetry and transitivity by constructing new cylindrical objects.

I would like to dispense with the notion of a non-functorial cylindrical object altogether. Instead, I am attracted by the search for natural operations on functorial cylindrical objects. For example, applying the cofibrate functorial factorization to the automorphism $\rm{swap}$ (of the codiagonal morphism) defines a natural automorphism of a cylinder, which gives a natural proof of the symmetry of cofibrate homotopy.

To prove transitivity, it is natural to glue two cylinders along the upper and lower bases and construct a natural isomorphism of the resulting 'long cylinder' $A \times 2I$ with the standard cylinder $A \times I$. It is easy to see that this is sufficient for gluing homotopies, i.e. transitivity.

In an arbitrary model category, such an isomorphism definitely cannot exist: say, in a trivial model structure $\rm{Cofib} = \cal{C}, \rm{Fib} = \rm{Iso}~\cal{C}, \rm{WeakEq} = \cal{C}$, it is easy to see that that $A \times I = A \amalg A, \; A \times 2I = A \amalg A \amalg A$ (although all cylindrical objects in any model category are weakly equivalent). Is there a natural wide class of model categories (possibly with an additional structure) in which it is possible to construct a natural isomorphism of $A \times I \to A \times 2I$ (at least for cofibrates $A$)? Or does the desire to limit ourselves to such categories select convenient model categories for some interesting homotopy categories? (I'm just starting to learn model categories)

In the excellent "A Handbook of Model Categories" (2021), cofibrant and fibrant homotopies are defined exactly as it seemed natural to me: immediately through functorial cylindrical objects / path space objects (hereinafter I will only talk about cylindrical objects), and the concept of non-functorial cylindrical objects in general not mentioned. But for proofs of elementary statements such as "cofibrate homotopy is an equivalence relation if dom is a cofibrant" Scott Balchin refers to Hovey, who proves symmetry and transitivity by constructing new cylindrical objects.

I would like to dispense with the notion of a non-functorial cylindrical object altogether. Instead, I am attracted by the search for natural operations on functorial cylindrical objects. For example, applying the cofibrate functorial factorization to the automorphism $\rm{swap}$ (of the codiagonal morphism) defines a natural automorphism of a cylinder, which gives a natural proof of the symmetry of cofibrate homotopy.

To prove transitivity, it is natural to glue two cylinders along the upper and lower bases and construct a morphism $\alpha \colon A \times I \to A \times 2I$ such that $i_0 \circ \alpha = i_0 \circ s_0$ and $i_1 \circ \alpha = i_1 \circ s_1$. Here $i_0, i_1 \colon A \to A \times I$ are compositions of inclusions $A \to A \amalg A$ with canonical embedding $A \amalg A \to A \times I$, and $s_0, s_1$ are push-out arrows.

Is there a natural wide class of model categories (possibly with an additional structure) in which it is possible to construct such morphism $A \times I \to A \times 2I$ (at least for cofibrates $A$)? Or does the desire to limit ourselves to such categories select convenient model categories for some interesting homotopy categories? (I'm just starting to learn model categories)

P.S. Of course, I write function composition in direct order (less common).

UPD. The question has been slightly edited, see version history for the context of Mike Shulman's answer.

In the excellent "A Handbook of Model Categories" (2021), cofibrant and fibrant homotopies are defined exactly as it seemed natural to me: immediately through functorial cylindrical objects / path space objects (hereinafter I will only talk about cylindrical objects), and the concept of non-functorial cylindrical objects in general not mentioned. But for proofs of elementary statements such as "cofibrate homotopy is an equivalence relation if dom is a cofibrant" Scott Balchin refers to Hovey, who proves symmetry and transitivity by constructing new cylindrical objects.

I would like to dispense with the notion of a non-functorial cylindrical object altogether. Instead, I am attracted by the search for natural operations on functorial cylindrical objects. For example, applying a cofibrate functorial factorization to an automorphism $\rm{swap}$ (of the codiagonal morphism) defines a natural automorphism of a cylinder, which gives a natural proof of the symmetry of cofibrate homotopy.

To prove transitivity, it is natural to glue two cylinders along the upper and lower bases and construct a natural isomorphism of the resulting 'long cylinder' $A \times 2I$ with the standard cylinder $A \times I$. It is easy to see that this is sufficient for gluing homotopies, i.e. transitivity.

In an arbitrary model category, such an isomorphism definitely cannot exist: say, in a trivial model structure $\rm{Cofib} = \cal{C}, \rm{Fib} = \rm{Iso}~\cal{C}, \rm{WeakEq} = \cal{C}$, it is easy to see that that $A \times I = A \amalg A, \; A \times 2I = A \amalg A \amalg A$ (although all cylindrical objects in any model category are weakly equivalent). Is there a natural wide class of model categories (possibly with an additional structure) in which it is possible to construct a natural isomorphism of $A \times I \to A \times 2I$ (at least for cofibrates $A$)? Or does the desire to limit ourselves to such categories select convenient model categories for some interesting homotopy categories? (I'm just starting to learn model categories)

In the excellent "A Handbook of Model Categories" (2021), cofibrant and fibrant homotopies are defined exactly as it seemed natural to me: immediately through functorial cylindrical objects / path space objects (hereinafter I will only talk about cylindrical objects), and the concept of non-functorial cylindrical objects in general not mentioned. But for proofs of elementary statements such as "cofibrate homotopy is an equivalence relation if dom is a cofibrant" Scott Balchin refers to Hovey, who proves symmetry and transitivity by constructing new cylindrical objects.

I would like to dispense with the notion of a non-functorial cylindrical object altogether. Instead, I am attracted by the search for natural operations on functorial cylindrical objects. For example, applying the cofibrate functorial factorization to the automorphism $\rm{swap}$ (of the codiagonal morphism) defines a natural automorphism of a cylinder, which gives a natural proof of the symmetry of cofibrate homotopy.

To prove transitivity, it is natural to glue two cylinders along the upper and lower bases and construct a natural isomorphism of the resulting 'long cylinder' $A \times 2I$ with the standard cylinder $A \times I$. It is easy to see that this is sufficient for gluing homotopies, i.e. transitivity.

In an arbitrary model category, such an isomorphism definitely cannot exist: say, in a trivial model structure $\rm{Cofib} = \cal{C}, \rm{Fib} = \rm{Iso}~\cal{C}, \rm{WeakEq} = \cal{C}$, it is easy to see that that $A \times I = A \amalg A, \; A \times 2I = A \amalg A \amalg A$ (although all cylindrical objects in any model category are weakly equivalent). Is there a natural wide class of model categories (possibly with an additional structure) in which it is possible to construct a natural isomorphism of $A \times I \to A \times 2I$ (at least for cofibrates $A$)? Or does the desire to limit ourselves to such categories select convenient model categories for some interesting homotopy categories? (I'm just starting to learn model categories)