Not every over-under-category is cocomplete

Question

Something is wrong between me and Hirschhorn: point 3 of this result (in the book Model categories and their localizations):

7.6.4. Homotopy in undercategories and overcategories.

Theorem 7.6.5. Let $\mathcal{M}$ be a model category.

1. If $A$ is an object of $\mathcal M$, then the category $(A \downarrow \mathcal M)$ of objects of $\mathcal M$ under A (see Definition 7.6.1) is a model category in which a map is a weak equivalence, fibration, or cofibration if it is one in $\mathcal M$.
2. If $X$ is an object of $\mathcal M$, then the category $(\mathcal M \downarrow X)$ of objects of $\mathcal M$ over $X$ ( Definition 7.6.2) is a model category in which a map is a weak equivalence, fibration, or cofibration if it is one in $\mathcal M$.
3. If $A$ and $B$ are objects in $\mathcal M$, then the category $(A \downarrow \mathcal M \downarrow B)$ of objects of $\mathcal M$ under $A$ and over $B$ (see Definition 7.6.3) is a model category in which a map is a weak equivalence, fibration, or cofibration if it is one in $\mathcal M$.

Pʀᴏᴏғ. This follows directly from the definitions. ❑

seems to be false taken as it is: either I misunderstood something, or $(A\downarrow \mathcal{M}\downarrow B)$ is seldom cocomplete (what should an initial object be?).

Is Hirschhorn wrong? Does he mean something different from the result he states?

Answer 1

The category $(A\downarrow \mathcal{M}\downarrow B)$ is not necessarily a model category, as it can fail to be complete or cocomplete (or both). For example, $(\{*\} \downarrow \mathsf{Set} \downarrow \varnothing)$ is not a model category: it is empty! There is no map from a singleton to the empty set.

However one can define, for every $f : A \to B$, the category: $$(A\downarrow \mathcal{M}\downarrow B)_f = \{ A \xrightarrow{g} X \xrightarrow{h} B \mid h \circ g = f \}$$ over objects over $B$ and under $A$ such that the morphisms compose to $f$. Then it's clear that $$(A\downarrow \mathcal{M}\downarrow B) = \bigsqcup_{f \in \hom_\mathcal{M}(A,B)} (A\downarrow \mathcal{M}\downarrow B)_f$$ is a disjoint union of categories (the different components don't interact at all), and each $(A\downarrow \mathcal{M}\downarrow B)_f$ is a model category with weak equivalences, fibrations and cofibrations as in Hirschhorn's definition (for example the initial object is $A \to A \xrightarrow{f} B$, the terminal object is $A \xrightarrow{f} B \to B$).