I am looking for references discussing the category $Filt$ of filters (in the sense of set theory, details below), its simplicial category $sFilt$ and its full subcategory $Top\hookrightarrow sFilt$ of topological spaces.

What is known about $sFilt$ and the embedding $Top\hookrightarrow sFilt$? Is there a model structure on $sFilt$ compatible with the embedding? Was it discussed in the literature? Is there an exposition of elementary topology, say within the first two chapters of Bourbaki, General topology, in terms of $sFilt$?

The only reference on the category of filters I was able to find is a paper of Andreas Blass, *Two closed categories of filters*,
Fund. Math. 94 1977 2 129 – 143. http://matwbn.icm.edu.pl/ksiazki/fm/fm94/fm94115.pdf, and references therein.
I found nothing on $sFilt$; of course, it is a subcategory of the category of simplicial topological spaces...

Now let me give some details and definitions.

Tho word "filter" is meant in the sense of set theory, i.e. a topological space such that any superset of a non-empty open subset is necessarily open (usually the space is assumed not to be discrete). By the category of filters I mean the full subcategory of $Top$ whose objects are topological spaces with this property.

The embedding $Top\hookrightarrow sFilt$ is straightforward, essentially the definition of a topological space in terms of a system of neighbourhoods. To a space $X$ there correspond the object $(|X|,|X|\times |X|, |X|\times |X|\times |X|,...)$ based of Cartesian powers of $|X|$ (set-wise) where (i) the topology on $|X|$ is antidiscrete, and (ii) a non-empty subset $U\subset |X|\times |X|$ is open iff for each $x\in X$ there is a neighbourhood $U_x\ni x$ such that $\{x\}\times U_x \subset U$, and (iii) the filter on $|X|^n$, $n>2$, is the coarsest filter such that all the maps $|X|^n\longrightarrow |X|^2,(x_1,...,x_n)\mapsto(x_i,x_{i+1})$, $1\leq i<n$ are continuous.

There is a similar embedding of uniform spaces (in the language of metric spaces, a non-empty subset of $|X|^n$ is open iff it contains an $\varepsilon$-neighbourhood of the diagonal for some $\varepsilon$)