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Stream: learning: questions

Topic: What's this operation on relations called?

Mike Stay (Nov 03 2020 at 19:50):

What's it called when $S(a,b) \Rightarrow T(f(a), g(b))?$ The relation $S$ doesn't imply $T$ directly, but this more restrictive implication. Does it have a name?

John Baez (Nov 03 2020 at 19:52):

If you had an "iff" I'd say the relation $S \subseteq X \times Y$ is the pullback of the relation $T \subseteq X' \times Y'$ along the map $f \times g : X \times Y \to X' \times Y'$.

John Baez (Nov 03 2020 at 19:53):

But since you just have an "if-then", I'd say $S$ implies the pullback $(f \times g)^\ast T$.

Mike Stay (Nov 03 2020 at 20:03):

Thanks! I guess it's a kind of morphism between spans where the feet don't have to match: not a map of spans or a span of spans, but the notion of morphism you'd get from the "multisorted theory of a span". The theory has three objects X, Y, Z and two morphisms $f:X\to Y$ and $g:X\to Z$. "Homomorphism" of spans? The relation case would be restricting to "jointly monic spans".

sarahzrf (Nov 05 2020 at 08:15):

that's a square in the double category of relations and functions

sarahzrf (Nov 05 2020 at 08:17):

(so, a special case of a square in the double category of spans)