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1. • CommentRowNumber1.
   • CommentAuthorporton
   • CommentTimeOct 17th 2014
   • PermaLink
   Author: porton
   Format: MarkdownItexFrom [topogeny](http://ncatlab.org/toddtrimble/published/topogeny): $Filt(X)$ is the filtered colimit completion (in the $2$-category of preorders or $\mathbf{2}$-enriched categories) of $P X$; otherwise put, $Filt(X) \cong Lex(P X, \mathbf{2})$. Please explain why filters are equivalent to $Lex(P X, \mathbf{2})$ to a category theory novice.

   From topogeny:

   $Filt(X)$ is the filtered colimit completion (in the $2$-category of preorders or $\mathbf{2}$-enriched categories) of $P X$; otherwise put, $Filt(X) \cong Lex(P X, \mathbf{2})$.

   Please explain why filters are equivalent to $Lex(P X, \mathbf{2})$ to a category theory novice.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 17th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexLet $Y$ be any meet-semilattice. (In the present case, $Y = P X$.) Then a poset map $\phi: Y \to \mathbf{2}$ is equivalent to an upward-closed subset $F$ of $Y$, by taking $F = \phi^{-1}(1)$. Having in addition $\phi(\top) = 1$ (so that $\phi$ preserves the empty meet) means $\top \in F$. Having in addition $\phi(y \wedge z) = \phi(y) \wedge \phi(z)$ means that $F = \phi^{-1}(1)$ is closed under meets. An upward-closed subset of $Y$ that contains $\top$ and is closed under meets is, by definition, a filter of $Y$.

   Let $Y$ be any meet-semilattice. (In the present case, $Y = P X$.) Then a poset map $\phi: Y \to \mathbf{2}$ is equivalent to an upward-closed subset $F$ of $Y$, by taking $F = \phi^{-1}(1)$. Having in addition $\phi(\top) = 1$ (so that $\phi$ preserves the empty meet) means $\top \in F$. Having in addition $\phi(y \wedge z) = \phi(y) \wedge \phi(z)$ means that $F = \phi^{-1}(1)$ is closed under meets. An upward-closed subset of $Y$ that contains $\top$ and is closed under meets is, by definition, a filter of $Y$.