Do pushouts of compactly generated Hausdorff spaces exist?

Question

Let $A\to X$ and $A\to Y$ be maps of compactly generated Hausdorff spaces. Does the pushout $X\coprod_A Y$ in the category of compactly generated Hausdorff spaces exist? If necessary, one can assume that both these maps are cofibrations.

Answer 1

Let $I$ be a small category and let $F:I\to\mathbf{CGHaus}$ be an $I$-shaped diagram of compactly generated Hausdorff spaces. Let us show that $F$ has a colimit in CGHaus. Let $C$ be its colimit in Top. Recall that $C$ may be constructed as the quotient $$ C=\coprod_{i\in I} Fi\Big/\sim $$ where $\sim$ is the equivalence relation generated by $x_i\sim Ff(x_i)$ for all $i\in I$ and $f$ morphism in $I$ with source $i$. As a quotient of (the disjoint union of compactly generated Hausdorff spaces, thus of) a compactly generated Hausdorff space, it may fail both to be compactly generated and Hausdorff. So let us instead consider $$C'=k\big(C^{\mathrm{sep}}\big)$$ the $k$-ification of the canonical Hausdorff quotient of $C$. By definition, $C'\in\mathbf{CGHaus}$. The functors $$\mathrm{sep}:\mathbf{Top}\longrightarrow\mathbf{Haus}\qquad\text{and}\qquad k:\mathbf{Top}\longrightarrow\mathbf{CG}$$ are both leftadjoint to the appropriate forgetful functor, so for any compactly generated Hausdorff space $X$, $$ \begin{align*} \mathrm{map}_{\mathbf{CGHaus}}(C',X)&=\mathrm{map}_{\mathbf{CGHaus}}(k(C^{\mathrm{sep}}),X)\\ &\simeq \mathrm{map}_{\mathbf{Top}}(C^{\mathrm{sep}},X)\\ &=\mathrm{map}_{\mathbf{Haus}}(C^{\mathrm{sep}},X)\\ &\simeq \mathrm{map}_{\mathbf{Top}}(C,X) \end{align*} $$ It is now obvious that $C'$ is a colimit in CGHaus to $F$. Indeed, any natural transformation $F\Rightarrow \mathrm{cst}_X$ from $F$ to the constant functor $X$ will give rise to a map $C\to X$ making the relevant diagrams commute, and thus to a map $C'\to X$ with the same property.

The case you were asking about is the special case where $I$ is the category $$ \begin{array}{ccc} \bullet & \longrightarrow & \bullet\\ \downarrow & & \\ \bullet & & \end{array} $$

Answer 2

Both categories $\mathbf{CG}$ and $\mathbf{Haus}$ have inclusion functors into $\mathbf{Top}$ which are part of an adjunction. While $\mathbf{CG}↪\mathbf{Top}$ is left adjoint to the functor $k$, with the counit being the identity map from the $k$-ification of $X$ to $X$ with its original topology, the inclusion $\mathbf{Haus}↪\mathbf{Top}$ is right adjoint to the functor $H$, the unit being the quotient map $X\twoheadrightarrow HX$ onto the largest Hausdorff quotient $HX$. These two categories are independent in the following sense:

• The reflector $H:\mathbf{Top}\to\mathbf{Haus}$ restricts to a reflector $\mathbf{CG}\to\mathbf{CGHaus}$ since each quotient of a compactly generated spaces is again compactly generated.
• The coreflector $k:\mathbf{Top}\to\mathbf{CG}$ restricts to a coreflector $k:\mathbf{Haus}\to\mathbf{CGHaus}$ since $k$ only refines the topology on a space.

The functor $H:\mathbf{CG}\to\mathbf{CGHaus}$ is a left adjoint, so it preserves colimits. That mean that for a diagram $D$ in $\mathbf{CGHaus}$ the colimit in $\mathbf{CGHaus}$ is the same as the Hausdorff quotient $H$ of the colimit in $\mathbf{CG}$, which coincides with the colimit of $D$ in $\mathbf{Top}$ as the inclusion $\mathbf{CG}↪\mathbf{Top}$ preserves colimits.

Dually, the functor $k:\mathbf{Haus}\to\mathbf{CGHaus}$ is a right adjoint, so it preserves limits. That means that for a diagram $D$ in $\mathbf{CGHaus}$ the limit in $\mathbf{CGHaus}$ is the same as the $k$-ification of the limit of $D$ in $\mathbf{Haus}$, which coincides with limit in $\mathbf{Top}$.

Since $\mathbf{Haus}$ is complete, so is $\mathbf{CGHaus}$. And since $\mathbf{CG}$ is cocomplete, so is $\mathbf{CGHaus}$.

So if you want the pushout in $\mathbf{CGHaus}$, then take the pushout in $\mathbf{Top}$ and apply the Hausdorff quotient $H$ to it.