1
$\begingroup$

Let $X$ be a poset.

Statement: A subset $Y\subseteq X$ is filtered if an only if there exists a filtered diagram (category) $D$ with a functor $D\rightarrow X$ such that the image of $D$ is $Y$.

How to prove this?

Can it be generalized?

$\endgroup$
2
$\begingroup$

The "only if" direction is easy: take $D = Y$ and the inclusion. For the "if" direction, just proceed directly: take two elements in $Y$; then they have a preimage in $D$, so they have an upper bound in $D$, so they have an upper bound in $Y$, etc.

Of course, one should note that filtered diagrams in the sense of category theory are upside down compared to filters in the sense of order theory.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.