I find that the "light cone" visualization helps to fix some of the inconsistencies:
This uses the Hasse diagram "smaller on bottom" order, but remains consistent with the standard symbols for "meet" and "join": the future light cone looks like a \(\vee\) for "least upper bound", and the past light cone looks like a \(\wedge\) for "greatest lower bound"). I still have to remember which symbol goes with which word, but things aren't quite so inconsistent from here.
(In other words, I don't interpret \(\vee\) and \(wedge\) as arrows, but as cones of light emanating from a flashlight. Think about the negative space around the symbol, rather than the symbol itself!)
I had a lousy time trying to memorize what made a "left adjoint" different from a "right adjoint".
This comes down to the defining property, where if "f" is a left adjoint to "g", then "f" appears on the left of the \(\le\): \(f(x) \le y \Leftrightarrow x \le g(y)\).
Comment Source:I find that the ["light cone"](https://en.wikipedia.org/wiki/Light_cone) visualization helps to fix some of the inconsistencies:
<center></center>
This uses the Hasse diagram "smaller on bottom" order, but remains consistent with the standard symbols for "meet" and "join": the future light cone looks like a \\(\vee\\) for "least upper bound", and the past light cone looks like a \\(\wedge\\) for "greatest lower bound"). I still have to remember which symbol goes with which word, but things aren't quite so inconsistent from here.
(In other words, I don't interpret \\(\vee\\) and \\(wedge\\) as arrows, but as cones of light emanating from a flashlight. Think about the negative space around the symbol, rather than the symbol itself!)
> I had a lousy time trying to memorize what made a "left adjoint" different from a "right adjoint".
This comes down to the defining property, where if "f" is a left adjoint to "g", then "f" appears on the _left_ of the \\(\le\\): \\(f(x) \le y \Leftrightarrow x \le g(y)\\).
Comments
I find that the "light cone" visualization helps to fix some of the inconsistencies:
This uses the Hasse diagram "smaller on bottom" order, but remains consistent with the standard symbols for "meet" and "join": the future light cone looks like a \(\vee\) for "least upper bound", and the past light cone looks like a \(\wedge\) for "greatest lower bound"). I still have to remember which symbol goes with which word, but things aren't quite so inconsistent from here.
(In other words, I don't interpret \(\vee\) and \(wedge\) as arrows, but as cones of light emanating from a flashlight. Think about the negative space around the symbol, rather than the symbol itself!)
This comes down to the defining property, where if "f" is a left adjoint to "g", then "f" appears on the left of the \(\le\): \(f(x) \le y \Leftrightarrow x \le g(y)\).
I find that the ["light cone"](https://en.wikipedia.org/wiki/Light_cone) visualization helps to fix some of the inconsistencies: <center></center> This uses the Hasse diagram "smaller on bottom" order, but remains consistent with the standard symbols for "meet" and "join": the future light cone looks like a \\(\vee\\) for "least upper bound", and the past light cone looks like a \\(\wedge\\) for "greatest lower bound"). I still have to remember which symbol goes with which word, but things aren't quite so inconsistent from here. (In other words, I don't interpret \\(\vee\\) and \\(wedge\\) as arrows, but as cones of light emanating from a flashlight. Think about the negative space around the symbol, rather than the symbol itself!) > I had a lousy time trying to memorize what made a "left adjoint" different from a "right adjoint". This comes down to the defining property, where if "f" is a left adjoint to "g", then "f" appears on the _left_ of the \\(\le\\): \\(f(x) \le y \Leftrightarrow x \le g(y)\\).