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Ruijun Lin's Space

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Connections between topological and categorical limits

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The definitions of a filter and its related concepts are from Tai-Danae Bradley, Tyler Bryson, and John Terilla, Topology: A Categorical Approach.

Definition. (filter) A filter on a set $X$ is a collection $\mathcal{F}$ that is (i) downward directed: $A,B\in \mathcal{F}$ implies there exists $C\in \mathcal{F}$ such that $C\subseteq A\cap B$, (ii) nonempty: $\mathcal{F} \neq \varnothing$, (iii) upward closed: $A\in \mathcal{F}$ and $A\subseteq B$ implies $B\in \mathcal{F}$. An additional property is often useful: (iv) proper: there exists $A\subseteq X$ such that $A\notin \mathcal{F}$.

Remarks:

  • Being downward directed and upward closed implies that filters are closed under finite intersections, and being proper is equivalent to the requirement that $\varnothing \notin \mathcal{F}$.
  • $2^{X}$ is itself a filter but not proper.
  • A set that is only downward directed and nonempty is called a filterbase. Any filterbase generates a filter simply by taking the upward closure of the base.

Examples.

  • (trivial filter) For any set $X$ there is the smallest filter ${X}$ called the trivial filter.
  • (eventuality filter) Given a sequence ${x_{n}}$ in a space $X$, the set

$\mathcal{E}_ {x_n} = \lbrace A \subseteq X \mid \text{ there exists an } N \text{ so that } x_n \in A \text{ for all } n \geq N \rbrace$

is a proper filter.

  • (non-example, open neighborhoods of a point $x$) Given a topological space $X$ and a point $x\in X$ , the collection of open neighborhoods of $x$ denoted by $\mathcal{T}_{x}$, is generally not a filter, but a filterbase.

Definition. (convergence) A filter $\mathcal{F}$ on a topological space $(X,\mathcal{T})$ converges to $x$ if and only if $\mathcal{F}$ refines $\mathcal{T}_x$, that is, if $\mathcal{T}_x \subseteq \mathcal{F}$. When $\mathcal{F}$ converges to $x$ we will write $\mathcal{F} \rightarrow x$.

As an easy observation, a sequence ${x_{n}}$ converges to a point $x$ if and only if $\mathcal{E}_{x_n}$.

With the concept of filters, we now give an equivalence statement between the categorical limit and the topological limit.

Let $X$ be a topological space and $\mathcal{F}_X$ be the collenction of all filters on $X$.

Given $x \in X$ and $F \in \mathcal{F}_{X}$, let $\mathcal{U}_X(x)$ be the neighborhood filter of $x$, which is the filter generated by $\mathcal{T}_x$ and let

$$ \mathcal{F}_{x,F} = \lbrace G \in \mathcal{F}_X \mid \mathcal{F} \cup \mathcal{U}_X(x) \subseteq G \rbrace $$

be the collection of all the filters containing both $F$ and the neighborhood filter of $x$.

We observed that $F \subseteq F \cup \mathcal{U}_ X(x) \subseteq \bigcap_{G\in \mathcal{F}_ {x,F}}G,$ and the latter two sets are filters.

Also since $\bigcap_ {G\in \mathcal{F}_ {x,F}}G$ is the smallest filter containing $F \cup \mathcal{U}_ X(x)$ and $F \cup \mathcal{U}_ X(x)$ is itself a filter, we have that $\bigcap_{G\in \mathcal{F}_ {x,F}}G \subseteq F \cup \mathcal{U}_ X(x)$ so that $\bigcap_ {G\in \mathcal{F}_ {x,F}}G = F \cup \mathcal{U}_ X(x)$.

View the inclusion of sets “$\subseteq$” as a partial order on $\mathcal{F}_ X$ and $\mathcal{F}_ {X,F}$. Then $\mathcal{F}_ X$ becomes a small category and $\mathcal{F}_ {X,F}$ is its subcategory. Let $E : \mathcal{F}_ {X,F} \to \mathcal{F}_X$ be the embedding functor. Now we have a theorem indicating the connection between the categorical limit and the topological limit.

Theorem. $x$ is a topological limit of the filter $F$ if and only if $F$ is a categorical limit of the functor $E$.

Proof. $x$ is a topological limit of the filter $F$, i.e., $F \to x$

$\Leftrightarrow (\mathcal{T}_x \subseteq \mathcal{U}(x)) \subseteq F \ \text{and} \ F \cup \mathcal{U}(x) \subseteq F$

$F = \bigcap_{G \in \mathcal{F}_{X,F}} g$ (the universal object)

$\Leftrightarrow (F \xrightarrow{p_{G}}E(G))_ {G \in \mathcal{F}_ {x,F}}$ is a limit cone of $E$

(i.e., $F$ is a categorical limit of $E$)

References

Updated on 2025-02-07
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