Definition 15.5.
A weak (co)limit of a diagram \(I \to \mathcal{C}\) is a cone (respectively, cocone) through which every other cone (respectively, cocone) factors, without requiring the factorization to be unique.
Section 15.2 Necessary Conditions and Weak Limits
A representable functor must convert colimits in \(\mathcal{C}\) into limits in \(\mathbf{Set}\text{.}\) We do not, however, need to check this on every kind of colimit, since some can be built from others (coequalizers from pushouts and coproducts; sequential colimits from coequalizers and coproducts).
The category we care about is \(\mathcal{C} = \mathrm{Ho}(\mathbf{Top}_*)\text{,}\) the homotopy category of pointed spaces. This category has coproducts but few other (strict) colimits — in particular, it does not have all pushouts. To work around this we relax the limit notions.
Example 15.6. Weak pullback.
A weak pullback of a cospan \(A \xrightarrow{f} C \xleftarrow{g} B\) is a commutative square
such that for every \(X\) with maps \(x \colon X \to A\) and \(y \colon X \to B\) satisfying \(fx = gy\text{,}\) there exists a (not necessarily unique) \(h \colon X \to P\) with \(ph = x\) and \(qh = y\text{.}\) If the strict pullback \(A \times_C B\) exists, this amounts to the canonical map \(P \to A \times_C B\) being a split epimorphism. In \(\mathbf{Set}\) this is just surjectivity.
Remark 15.7.
Representable functors send weak colimits in \(\mathcal{C}\) to weak limits in \(\mathbf{Set}\text{.}\)
The homotopy extension property (Definition 7.10) lets us glue maps that agree only up to homotopy. Consequently, a pushout along a cofibration gives a weak pushout in the based homotopy category. Concretely, if
is a pushout in \(\mathbf{Top}_*\) and \(A \to X\) is a cofibration, then this square is a weak pushout in \(\mathrm{Ho}(\mathbf{Top}_*)\text{.}\)
