How do you know when a reflective subcategory of Top is quotient-reflective?

Question

A subcategory $\mathcal{C}$ of the category $Top$ of topological spaces is a reflective subcategory if the inclusion functor $i:\mathcal{C}\hookrightarrow Top$ has a left adjoint $R:Top\rightarrow \mathcal{C}$. In other words, for every space $X$, there is a space $RX\in \mathcal{C}$ and a map $r:X\rightarrow RX$ such that for each map $f:X \rightarrow Y$ with $Y\in \mathcal{C}$, there is a unique map $\tilde{f}:RX\rightarrow Y$ such that $\tilde{f}\circ r=f$.

If the reflection map $r$ is always a quotient map, $\mathcal{C}$ is said to be a quotient-reflective subcategory. For example, it seems that the subcategories of $T_0$ spaces and $T_2$ spaces are quotient-reflective but the subcategories of completely regular spaces and compact Hausdorff spaces are reflective but not quotient-reflective. I'd like to hear of other examples of quotient-reflective subcategories as well.

There seem to be conditions fully characterizing when a subcategory is reflective. For example, some necessary conditions are mentioned in the answer to this MO question. Are there also conditions characterizing subcategories which are quotient-reflective?

Answer 1

The following theorem (16.8 in Abstract and Concrete Categories, Adamek-Herrlich-Strecker) could be of use.

Let E and M be subclasses of epis and monoes respectively, closed under composition with isomorphisms. If A is a full subcategory of an (E,M)-factorisable category B, then the following conditions are equivalent:

(1) A is E-reflective in B.

(2) A is closed under the formation of M-sources in B.

In the case that B has products and is E-co-wellpowered, the above conditions are equivalent to:

(3) A is closed under the formation of products and M-subobjects in B.

In the case of TOP, E would be extremal epis, and M monoes. In topological constructs extremal epis are exactly quotients (final and surjective). TOP is extremal epi-mono factorisable, has products and is extremally co-wellpowered. So a full subcategory is quotient reflective in TOP if and only if it is closed under products and (not necessarily initial) subobjects.

The subcategories of completely regular spaces is for example not closed for "adding more opens", so it is, as you said, not quotient reflective.

Answer 2

If you'll allow me to abstract away from $Top$ for a minute, there are many examples of quotient-reflective subcategories. A simple example is that abelian groups are quotient-reflective in groups, because the unit of the adjunction is a quotient map $\pi: G \to G/[G, G]$. In general, if you start with an algebraic theory $T$ whose signature is given by a set of function symbols, and if $T'$ has the same signature but more universally quantified equational axioms in addition to those already in $T$, then the category of $T'$-models, as a full subcategory of the category of $T$-models, is quotient-reflective: if $G$ is a $T$-model, then the unit of the adjunction is the quotient of $G$ by the smallest $T$-closed congruence which contains pairs of terms whose equality is asserted in $T'$.

A slightly different example is that sheaves (w.r.t. any site) are quotient-reflective in the category of presheaves on the underlying category of the site. (Actually, this may be somewhat similar, as one sheafifies by applying the plus construction twice, and applying the plus construction once reflects into separated presheaves, which are defined by extra equations on presheaves.)

The examples you give in $Top$ are interesting: in each of the $T_0$ and $T_2$ cases, the extra condition that defines the full subcategory asserts an equational conclusion. Thus, $T_0$ says $x = y$ if $x$ and $y$ have the same neighborhoods; $T_2$ says that $x = y$ if every open of $(x, y)$ in $X \times X$ contains a point $(z, z)$. They do not assert an existential conclusion on points as say compactness would (e.g., every ultrafilter converges to some point), which would require adjoining extra points when one passes to the reflection, in order to witness such existence. So it seems a useful criterion for quotient-reflectivity for concrete categories of models is that the full subcategory is defined by extra universally quantified equational conditions in the language used to specify the ambient category, just as in the algebraic examples above.

(By the way, I am intepreting "quotient" to mean "regular epi", i.e., a map that is the coequalizer of its own kernel pair, and I interpret "subcategory" here to mean "full subcategory, as is usual when discussing reflective subcategories. I'm not sure I can give a more abstract nonsense reply right now, but this should give a useful class of examples.)