2 typos

Nice observation. A sufficient condition so that the $2$-categoricall pull-back coincides with the strict one is that either $f\colon A\rightarrow C$ or $g\colon A\rightarrow C$ is a fibration, in the sense that it satisfies the isomorphism lifting property. These are fibrations for a model category structure on small categories with equivalences of categories as weak equivalences.

Let me recall the isomorphism lifting property. The functor $f$ satisfies that property if given an isomorphism $\epsilon\colon f(a)\rightarrow c$ in $C$ there exists an isomorphism $\delta\colon a\rightarrow b$ in $A$ with $f(\delta)=\epsilon$. In particular $f(b)=c$.

If $f$ satisfies this property then the fully-faithful functor $$A\times_CB\longrightarrow A\times^2_CB$$ $$(a,b)\mapsto (a,\operatorname{id},b)$$ is also esentially essentially surjective, hence an equivalence of categories. Indeed, given $(a,\epsilon, b)$ in the target, $\epsilon$ an isomorphism $\epsilon\colon f(a)\rightarrow g(b)$, we take an isomorphism $\delta\colon a\rightarrow b'$ with $f(\delta)=\epsilon$, and then $(\delta, \operatorname{id})\colon (a,\epsilon, b)\rightarrow (b',\operatorname{id},b)$ is an isomorphism.

If I remember well Waldhausen's paper, this condition applies whenever he invokes the apparently wrong conditiocondition.

1

Nice observation. A sufficient condition so that the $2$-categoricall pull-back coincides with the strict one is that either $f\colon A\rightarrow C$ or $g\colon A\rightarrow C$ is a fibration, in the sense that it satisfies the isomorphism lifting property. These are fibrations for a model category structure on small categories with equivalences of categories as weak equivalences.

Let me recall the isomorphism lifting property. The functor $f$ satisfies that property if given an isomorphism $\epsilon\colon f(a)\rightarrow c$ in $C$ there exists an isomorphism $\delta\colon a\rightarrow b$ in $A$ with $f(\delta)=\epsilon$. In particular $f(b)=c$.

If $f$ satisfies this property then the fully-faithful functor $$A\times_CB\longrightarrow A\times^2_CB$$ $$(a,b)\mapsto (a,\operatorname{id},b)$$ is also esentially surjective, hence an equivalence of categories. Indeed, given $(a,\epsilon, b)$ in the target, $\epsilon$ an isomorphism $\epsilon\colon f(a)\rightarrow g(b)$, we take an isomorphism $\delta\colon a\rightarrow b'$ with $f(\delta)=\epsilon$, and then $(\delta, \operatorname{id})\colon (a,\epsilon, b)\rightarrow (b',\operatorname{id},b)$ is an isomorphism.

If I remember well Waldhausen's paper, this condition applies whenever he invokes the apparently wrong conditio