ScalingStacks

Example 1.6. The forgetful functor

is a cartesian fibration. Given a morphism $U\rightarrow Y$ in ${\mathcal{S}\textup{et}}$ and a topological space $\mathcal{Y}\in{\mathcal{T}\textup{op}}$ equipped with an isomorphism $\textup{U}_{\mathcal{T}\textup{op}}(\mathcal{Y})\cong Y$ in ${\mathcal{S}\textup{et}}$, a $\textup{U}_{\mathcal{T}\textup{op}}$-cartesian lift is provided by endowing the set $U\in{\mathcal{S}\textup{et}}$ with the induced topology: this yields a topological space $\mathcal{U}\in{\mathcal{T}\textup{op}}$ equipped with a map $\mathcal{U}\rightarrow\mathcal{Y}$ in ${\mathcal{T}\textup{op}}$ and an isomorphism $\textup{U}_{\mathcal{T}\textup{op}}(\mathcal{U})\cong U$ in ${\mathcal{S}\textup{et}}$, which has the universal property that for any $\mathcal{Z}\in{\mathcal{T}\textup{op}}$ with underlying set $Z=\textup{U}_{\mathcal{T}\textup{op}}(\mathcal{Z})\in{\mathcal{S}\textup{et}}$, the resulting diagram

is a pullback square in ${\mathcal{S}\textup{et}}$. (If the morphism $U\rightarrow Y$ is actually the inclusion of a subset, this specializes to define the subspace topology on the set $U$.)