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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 $\varepsilon$)

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  • $\begingroup$ For a work before Blass see the paper by Koubek and Reiterman, for a later use of it - the paper by Moerdijk, for more general construction see nLab entry on the topos of types and references therein, $\endgroup$ – მამუკა ჯიბლაძე Nov 21 '17 at 21:28
  • $\begingroup$ as well as references to several works by Palmgren in the subsection "Sheaves on the topos of filters" of the nLab entry for nonstandard analysis. As for the category of simplicial filters - why do you think anybody has considered it before? The embedding of topological spaces and uniform spaces to it is your discovery, is not it? By the way, is it easy to see that this is indeed an embedding? Is it also full? Do you have it published somewhere? $\endgroup$ – მამუკა ჯიბლაძე Nov 21 '17 at 21:36
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    $\begingroup$ thanks. People consider all sorts of simplicial things, why not simplicial filters? Yes, it is a fully faithful functor, and verification is straightforward if tedious. I have not seen this construction explicitly in the literature; it appears in a research proposal draft,p.2 $\endgroup$ – user117509 Nov 21 '17 at 22:50
  • $\begingroup$ Thanks, it is very interesting. I believe you can safely assume that nobody considered it before. Are you sure you need the whole $\Delta$ to have a full embedding? Seems like you need very few of $\Delta$s morphisms for that... $\endgroup$ – მამუკა ჯიბლაძე Nov 22 '17 at 1:18
  • $\begingroup$ That's right, to have a full embedding you only need only two "first" objects of $\Delta$, namely $\{0\}$ and $\{0,1\}$. $\endgroup$ – user117509 Nov 22 '17 at 9:13

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