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have adjusted the section-outline
am hereby moving the following ancient query-box discussion from the entry to here:
+– {: .query} Mike Shulman: Is that really the right definition? I think of “strictly increasing” as meaning that $x \lt y$ implies $f(x) \lt f(y)$, which is equivalent to the above for linear orders but weaker for partial orders. But I don’t have much experience with strictly increasing functions between non-linear orders, so maybe that is the right definition for partial orders.
However, I don’t think it is the right definition for preorders; among other things, it’s not invariant under equivalence of categories. It seems to me that what you really want to say is that it is pseudomonic as a functor (whereas my weaker definition would become the statement that it is conservative as a functor.)
Toby: This is the definition in HAF (Section 3.17), which defines it for posets (and is a smart enough book that it wouldn't blindly extend a definition from a special case). Although I don't have a reference, I'm pretty sure that this also used in analysis and topology when thinking about convergence and nets, where they may be prosets. However, I think that you have a good point about preordered sets, so I've changed the wording above. (I'll also try to confirm how covergence theorists define ’strictly increasing’ functions between directed prosets.)
It occurs to me that, in the absence of the axiom of choice, one ought to accept even anafunctors between prosets as morphisms, even though these may not be representable as strict functions at all. I'll save that for another day, however.
Mike Shulman: Of course, the definition you gave above isn’t the same as pseudomonic unless $T$ is a partial order; in general you want to say $x\leq y$ whenever $f(x) \cong f(y)$. The version with $=$ is still not invariant under equivalence of $T$.
I don’t know a whole lot about convergence and nets, but I don’t remember seeing strictly increasing functions used there; I look forward to seeing what you find. Does HAF use the poset version for any application that makes clear why this is a good definition? Of course, monomorphisms of posets may quite naturally something to be interested in, but the question is why they should be called “strictly increasing.” =–
have adjusted the wording in the Idea-section
have put a Remark-environment around this remark:
Sometimes the term ’monotone’ or ’isotone’ (but rarely both) is used for functions from $S$ to itself such that
$x \leq f(x)$for all $x$ in $S$.
am hereby removing the following box comment that went with this:
+– {: .query} Is there a widely accepted term for this? I've seen both of these, I think, but the other meaning seems to be more common for both. —Toby =–
For what it’s worth, I have not seen either of these uses and I doubt that it’s helpful. I vote for removing the above remark altogether, or else add a compelling reference that uses terminology in this way.
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