Cube Lemma on a cofibrantly generated (almost) model category

Question

Suppose I have a complete and cocomplete category $\mathscr{C}$ with two sets of maps $I,J$ that are the candidates for generating (trivial) cofibrations on a model structure on $\mathscr{C}$. The only property I'm left to check in order to apply the recognition principle for cofibrantly generated model categories is that pushouts of maps in $J$ are weak equivalences.

In this framework, does the cube lemma hold already?

More precisely, if I have two spans $B_i \leftarrow A_i \rightarrow C_i$ for $i \in \{1,2\}$, where all the objects are cofibrant and $A_i \rightarrow B_i$ is a cofibration $\forall \ i \in \{1,2\}$, plus I'm given a pointwise weak equivalence of spans, is the induced map $$B_1 \coprod_{A_1} C_1 \to B_2 \coprod_{A_2} C_2$$ a weak equivalence as well?

Thanks in advance for any hint/reply.

Answer 1

If I understand your situation correctly, the answer is yes, the cube lemma holds, but you cannot use that to get a full model structure. Your situation often arises when trying to transfer a model structure from a cofibrantly generated model category $M$, along an adjunction $F:M\leftrightarrows N: U$, to a bicomplete category $N$, e.g. to a category $N$ of algebras over a monad. Check out Lemma 2.3 in Schwede-Shipley Algebras and Modules in monoidal model categories. For them also, the only difficulty is proving that the pushout is a trivial cofibration in $N$ (defined via $J$-cell) is a weak equivalence. The good news is that, even without this condition, you can probably prove whatever you want. Specifically, the structure on $N$ may be that of a semi-model structure. Have a look at Definition 2.3 in Spitzweck's thesis. Note that on page 14 he remarks that the cube lemma holds in this setting. Indeed, Karol's answer is spot on: in a semi-model category, pushouts of spans $A\gets B \to C$, where $B \to C$ is a trivial cofibration and $A$ is cofibrant, yield trivial cofibrations.

Even if your $N$ is not arising via a transfer along an adjunction, it may still be a semi-model category as defined by Benoit Fresse in 12.2.1 of this book. Again, the cube lemma holds. In general, any statement about model categories holds for semi-model categories if you restrict to the subcategory of cofibrant objects, or cofibrantly replace everything in sight. Since the cube lemma is already about cofibrant objects, there is no problem.

Unfortunately, it does NOT follow that every semi-model category is a model category. An example is the category of non-reduced symmetric operads in $M = Ch(\mathbb{F}_2)$. As a category of algebras over a $\Sigma$-cofibrant colored operad $P$, it is a semi-model category, by Spitzweck's Theorem or by Fresse 12.2.A applied to the transfer from the category of collections $\prod_{n \in \mathbb{N}} M^{\Sigma_n}$. However, it is not a full model structure: the pushout of $P(0)\to P(K)$, where $K$ is an acyclic chain complex $C$ in level zero and 0 otherwise, along the map $P(0)\to Com$ to the terminal non-reduced symmetric operad, is not a weak equivalence, because it introduces a summand of $(C\otimes C) / \Sigma_2$.

My advice: stick with the semi-model structure and use it to prove whatever you need. This was the approach taken by Spitzweck, by Goerss-Hopkins in their obstruction theory paper, by Fresse, by Hovey in his paper Monoidal Model Categories, and in many of my papers. A semi-model structure is good enough.

Answer 2

This is not an answer in a full generality, but it is certainly not the case if the domains of generating acyclic cofibrations are cofibrant. (I have a hard time thinking of an example of a model category where this is not true, but there are probably some.)

Assuming that the "Cube Lemma" holds (it's more often called the Gluing Lemma), take $A_2 \to B_2$ to be any generating acyclic cofibration and let $A_2 \to C_2$ be a morphism to a cofibrant object. Set $A_1 = B_1 = A_2$ and $C_1 = C_2$ and fill the cube with obvious maps. Then the Gluing Lemma says that the pushout of $A_2 \to B_2$ along $A_2 \to C_2$ is a weak equivalence.