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Stream: learning: questions

Topic: Externalizing local properties

John Onstead (Jul 17 2025 at 11:22):

Let $P\!\to\!\mathrm{Top}$ be a full subcategory defining a property $p$. Then a space $X$ is locally $p$ if each $x\in X$ has a neighborhood $U_x\subset X$ (with the subspace topology) lying in $P$. This works well in $\mathrm{Top}$ but fails to generalize beyond “nicely concrete” categories over $\mathrm{Top}$ (e.g. topological groups or smooth manifolds).

John Onstead (Jul 17 2025 at 11:23):

To address this, I've come up with what I think is a good way to externalize this notion to work in any category. Given the setup above, say $X$ is locally $p$ if there exists a jointly effective (or regular) epic sink $\{f_i:U_i\to X\}$ with each $U_i\in P$. This makes sense in any category. I'll consider this notion a "good externalization" precisely when, for every $p$, the locally‑$p$ spaces in $\mathrm{Top}$ under the classical definition form a full subcategory of those under the epic sink definition. That way, any topological "local property" is simply a "sink-based local property" perhaps satisfying some extra conditions.

John Onstead (Jul 17 2025 at 11:24):

My question: Is my notion of local property a good externalization? Equivalently, is every sink consisting of a choice of neighborhoods $U_x \in P$ for $x \in X$ induced by some jointly effective epi sink $\{V_j\to X\}$ with all $V_j\in P$? This is of course automatic if each of the neighborhoods were an open neighborhood (since an open cover is joint effective), but I'm not sure if it's sufficient to only consider open neighborhoods.

Morgan Rogers (he/him) (Jul 17 2025 at 11:39):

It really depends on P, I would think!

Mike Shulman (Jul 17 2025 at 15:17):

I think this is always true, since covers by not-necessarily-open neighborhoods should also be effective-epi. You can check this by hand, or use the fact that open covers are universally effective-epi, and universally effective-epi sinks generate a Grothendieck topology, hence satisfy saturation condition (3) here.

John Onstead (Jul 17 2025 at 19:58):

Mike Shulman said:

or use the fact that open covers are universally effective-epi, and universally effective-epi sinks generate a Grothendieck topology, hence satisfy saturation condition (3) here.

Let me see if I understand correctly... if we have a sink $U_i$ and a sink $V_j$, and $U_i$ is a covering family in a Grothendieck topology, then including $V_j$ in the topology won't change the category of sheaves if each morphism in $U_i$ factors through some morphism in $V_j$. So applied to the standard topology on $\mathrm{Top}$, I'm guessing we can take this to imply that if each open in some open cover factors through some morphism in another sink $V_j$, then $V_j$ is effective epi as well. (I think it does since if $V_j$ weren't joint effective epi, including it in the topology would make it no longer subcanonical and therefore would necessarily change the category of sheaves; since including it doesn't change the sheaves, it therefore implies it must be joint effective epi)

John Onstead (Jul 17 2025 at 19:58):

Then we can use the fact that a neighborhood of a point is defined to be a subset of a topological space that contains some open set containing that point. So if $N_x$ is the neighborhood for a point that contains an open set $U$, then the inclusion of $U$ into the space will factor through the inclusion of $N_x$. So now if we let $V_j$ be a selection of a neighborhood for every point in our space, we can let $U_i$ be a selection of an open set contained by each neighborhood in $V_j$, and indeed each morphism in $U_i$ factors through one in $V_j$. So by the above reasoning this should make $V_j$ effective epi.

Mike Shulman (Jul 17 2025 at 20:49):

Yep.