geometry and (co)limits
You can study geometry via analytical techniques and take the continuum as your unit of investigation. The continuum is of course a structure that is created by using set theoretical limit processes.
You can study geometry via algebraic techniques and take simplices as your unit of investigation. This time, not your unit of investigation, but your investigation techniques will involve categorical colimit processes.
The study of geometry can only be conducted via infinitary processes. You can not escape from this fact which stems from the inherent malleability of the notion of space.
Preferring the categorical colimit processes which work at the technique level has one important advantage though: It allows you to enlarge the scope of applicability of your investigations by keeping your unit of investigation as general as possible. If you are going to do something monstrous, do it on yourself not on your subject.
You can study geometry via algebraic techniques and take simplices as your unit of investigation. This time, not your unit of investigation, but your investigation techniques will involve categorical colimit processes.
The study of geometry can only be conducted via infinitary processes. You can not escape from this fact which stems from the inherent malleability of the notion of space.
Preferring the categorical colimit processes which work at the technique level has one important advantage though: It allows you to enlarge the scope of applicability of your investigations by keeping your unit of investigation as general as possible. If you are going to do something monstrous, do it on yourself not on your subject.
