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Stream: learning: questions

Topic: topological space is arrow from lattice to image of powerset

Ignat Insarov (Jul 11 2025 at 04:36):

Is it fair to say that a topological space is an arrow from a lattice (with some additional limits) to the image of the contravariant power set functor? Then, the category of topological spaces and continuous functions is the category of arrows to $\mathcal P$ and commuting squares in the ambient category of lattices (with some additional limits), like so:

$$\begin {array} {ccc} \\ L & \overset m \longleftarrow & K \\ s \Big \downarrow {\phantom s} & \overset {⟨f; m⟩} \longrightarrow & {\phantom t} \Big \downarrow t \\ \mathcal P (A) & \overset {\mathcal P f} \longleftarrow & \mathcal P (B) \end {array}$$

Here, $s$ and $t$ are two topological spaces and $⟨f; m⟩: s \rightarrow t$ is a continuous function.

If this is true, I should expect this definition to appear in the first chapter of any introduction to topology, but I do not think I have ever seen it out there. Is my definition wrong?

Kevin Carlson (Jul 11 2025 at 05:09):

I don’t quite understand but I’m not sure if it’s just a superficial consequence of your word choices. What is the lattice, here? The lattice of opens of the space? And you want a function from this lattice to some powerset? Are you saying you want the topology to be a function $T\to P(X)$ rather than a subset of $P(X)$? You could try that, but then “the same” open set can have multiple “names” in $T.$ This is a reasonable sounding thing to do, for a category theorist, but would be hard to explain on the first page of a topology text. But maybe I’ve totally misread you?

Ryan Wisnesky (Jul 11 2025 at 05:16):

In "point-free" topology they throw away sets of points and instead focus on the lattice formed by the open subsets, but the resulting subject isn't quite the same as point-set topology. Maybe you are noticing that much but not all information about a topological space is related to the algebraic properties of the lattice those spaces form?

Chris Grossack (she/they) (Jul 11 2025 at 05:37):

In Chapter 5 of Vickers's "Topology Via Logic" he defines a topological system to be a relation $\vDash$ between a set $X$ (thought of as points) and a frame $A$ (thought of as opens). Iirc he explicitly says these are the same thing as frame maps $A \to \mathcal{P}(X)$, and compares this with the classical notion of a topology on $X$. As Kevin said, they can't be exactly the same, because a frame map might not be injective, so that multiple elements of $A$ give the same set of points of $X$.

Ignat Insarov (Jul 11 2025 at 07:37):

What if we say «a topological space is an equivalence class of monic arrows» instead of simply «an arrow»? Since lattices (even with some additional limits) are algebraic theories, monic arrows should simply be defined by one to one functions (that respect the algebraic structure). This makes the definition of a topological space very similar to the definition of a subset, which is an equivalence class of monic arrows in the category of sets and functions.

@Kevin Carlson Yes, this is what I had in mind.
@Chris Grossack (she/they) A book I did not know I need!

P. S. It turns out that even in concrete categories there can exist monic arrows other than one to one functions. This needs to be looked into, but I am out of time.

Kevin Carlson (Jul 11 2025 at 16:17):

An equivalence class of monic arrows is indeed nothing more than a subset (in this case: you haven’t set us in any category other than the category of sets), and “a subset of some $P(X)$” (closed under the usual conditions) is just the usual definition of topology.