# How to get by with only functorial cylindrical objects?

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.

• It seems to me your real objection is not non-functoriality but rather non-uniqueness. Put it another way, it sounds to me you want to find a way to choose just one cylinder object (for each object) once and for all. I think this is not always possible. Certainly standard practice in model category theory is to allow oneself to choose ad hoc cylinder objects as required for the argument. Aug 7 at 14:53
• By functorial cylindrical object I mean the cylindrical object resulting from the factorization of the codiagonal map. I want to fix it as the meaning of the word cylinder, in particular, it is unique (and functorial), yes. But no other choice (once and for all) of a cylindrical object also seems natural to me. Aug 7 at 16:10

You might be interested in looking into enriched model categories. If $$\mathcal{V}$$ is a monoidal model category (with cofibrant unit, for simplicity) and $$\mathcal{C}$$ is a $$\mathcal{V}$$-model category, then for any interval object (i.e. cylinder object for the unit object) $$\mathbb{1}+\mathbb{1} \to \mathsf{I} \to \mathbb{1}$$ in $$\mathcal{V}$$, the copower (tensor) $$(-\odot \mathsf{I})$$ is a well-behaved "cylinder functor" on $$\mathcal{C}$$. I put it in quotes because in general $$A\odot \mathsf{I}$$ may not be a "cylinder object" in the model categorical sense, but it is whenever $$A$$ is cofibrant, which is usually enough.
In this case, there really is an object $$2\mathsf{I} = \mathsf{I} \sqcup_{\mathbb{1}} \mathsf{I}$$ such that your "long cylinder" is also an enriched copower $$A \odot 2\mathsf{I}$$. So what you want is an isomorphism $$\mathsf{I} \cong 2\mathsf{I}$$ in $$\mathcal{V}$$, or at least a morphism $$\mathsf{I} \to 2\mathsf{I}$$ that you can compose with to implement transitivity.
Whether such a thing exists depends of course on $$\mathcal{V}$$. The most common choice of $$\mathcal{V}$$ is simplicial sets, in which case such an (iso)morphism does not exist. But another traditionally important choice of $$\mathcal{V}$$ (though it's fallen somewhat out of favor recently) is topological spaces, and in this case there really is an isomorphism $$\mathsf{I} \cong 2\mathsf{I}$$. So you may be interested in looking into topologically enriched model categories.
• Model categories of chain complexes, on the other hand, are generally enriched over a monoidal model category of chain complexes, and I believe the latter does have a morphism (though not an isomorphism) $\mathsf{I} \to 2\mathsf{I}$ that can implement transitivity. Aug 9 at 17:43
• Indeed, what is really needed is only a morphism $\alpha A \times I \to A \times 2I$ such that $i_0 \circ \alpha i_0 \circ s_0$ and $i_1 \circ \alpha = i_1 \circ s_1$. Here $i_0, i_1 \colon A \to A \times I$ compositions of inclusions $A \to A \amalg A$ with canonical embedding $A \amalg A \to A \times I$, and $s_0, s_1$ push- out arrows. In my example with a trivial model structure, such a morphism is $A \amalg A \to A \amalg A \amalg A$ (embeddings $1 \mapsto 1, 2 \mapsto 3$). I'll edit the question accordingly, thank you! Aug 9 at 20:20