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The category of topological spaces fully faithfully embeds into the category of simplicial objects of the category of filters, where “the category of filters” means the full subcategory of topological spaces consisting of spaces such that any superset of a non-empty open subset is open (details below).
The embedding is essentially the definition of a topological space in terms of neighbourhood systems, e.g. (Bourbaki, General Topology, $I\S1.2,Ax.(V_I)-(V_IV)$).
Do you know any references this embedding or the category of simplicial objects in the category of filters, perhaps under another name? I was not able to find any; nlab does not seem to have an entry for either the category of filters or its simplicial category.
I sketch the construction below and a couple of open questions it leads to. More details appear in a research proposal [1], along with a number of questions. Little attempt is made to answer these questions, as probably they are already well-known.
The embedding $t:Top\longrightarrow sFilt$ into the category $sFilt$ of simplicial objects in the category $Filt$ of filters:
set-wise a space X is sent to $(|X|, |X|\times |X|,|X|\times |X|\times |X|,....)$,
where $|X|$ is the set of points of $X$
connecting maps are coordinate maps,
filters on $|X|^n$ are defined as follows:
a subset $U$ of $|X|$ is big iff $U=X$
a subset $U$ of $|X|\times |X|$ is big iff each point $x$ of $X$ has an open neighbourhood $U_x$ such that $\{x\}\times U_x \subset U$.
for $n \ge 3$, the filter on $|X|^n$ is the coarsest filter such that all coordinate maps $|X|^n-\longrightarrow |X|\times |X|$ are continuous wrt the topology on $|X|\times |X|$ defined above.
There is a similar but simpler embedding of metric spaces with uniformly continuous maps: a subset of $|X|^n$ is big iff it contains an $\epsilon$-neighbourhood of the diagonal for some $\epsilon$.
Two open questions:
Is there a model structure on the larger category compatible with a model structure on topological spaces?
Topological spaces, metric (uniform) spaces and filters all “live” in the same larger category, and this allows to use language of category theory to talk about compactness, precompactness, completeness, and equicontinuity. Can one reformulate and prove Arzela-Ascoli theorem category-theoretically?
Here is an example of use of the language.
A sequence $f_i:X\longrightarrow M$ of functions from a topological space $X$ to a metric space $M$ is equicontinuous iff the following morphism is well-defined:
$i(\{\omega\}) \times t(X) \longrightarrow m(M)$ $(i,x) \mapsto f_i(x)$
A sequence $f_i:X\longrightarrow M$ of functions from a metric space $X$ to a metric space $M$ is uniformly equicontinuous iff the following morphism is well-defined:
$i(\{\omega\}) \times m(X) \longrightarrow m(M)$ $(i,x) \mapsto f_i(x)$
A sequence $f_i:X\longrightarrow M$ of functions from a metric space $X$ to a metric space $M$ is uniformly Cauchy iff the following morphism is well-defined:
$E(\omega_{cofinite}) \times m(X) \longrightarrow m(M)$ $(i,x) \mapsto f_i(x)$
Here $t:Top\longrightarrow sFilt$, $m:UniformSpaces\longrightarrow sFilt$ are the two embeddings mentioned above into the category $sFilt$ of simplicial objects in the category $Filt$ of filters;
$\omega_{cofinite}$ is the filter of cofinite subsets of $\omega$; $\{\omega\}$ is the filter on $\omega$ with only one big subset $\omega$ itself;
the functor $i:Filt\longrightarrow sFilt$, $F\mapsto (F,F,F,...)$ with connecting maps being always identity, the functor $E:Filt\longrightarrow sFilt$, $F\mapsto (F,F\times F,F\times F\times F,...)$ sends a filter into the sequence of its Cartesian powers with coordinate maps.
References:
[1] Topological and metric spaces as full subcategories of the category of simplicial objects of the category of filters. A draft of a research proposal
Well, at least the “definition of a topological space in terms of neighbourhood systems” is at pretopological space.
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