When stating the existence of universal objects in mathematics, one often has to write something like:
For every object X there exist an object Y and a map f : Y -> X such that […] holds.
Here, both Y and the map f : Y -> X are part of the data so to speak. The object Y would be useless without the map f.
Question: Is the usage of there exist correct, or should it be there exists? I very often see the second one in textbooks, but I think the first one is correct. Does it change anything if I use a comma before 'and'?
Best Answer
You want to know whether in this case the conjunction "and" identifies a singular object composed of two parts, or two objects which are therefore plural.
As a hint:
Example: Bacon and Eggs
There is no bacon, and there are no eggs either.
Here there is potentially a different situation. Perhaps eggs are available on their own.
So "Bacon and Eggs" can be either singular or plural, but the meaning is not the same.
Example: there exist an object Y and a map ...
Here, the situation is the same. The same objects exist, with the same relation. It's meaningless to consider the map in the absence of Y since Y is part of the map, so neither version would be unclear. It's up to you which you use.
However if it sounds awkward, you can improve it by substituting "having" for "and".