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The minute before I had entered offline territory a few days ago, I had expanded the list of examples of (commuting) diagrams at diagram.
I added a few more, and also improved (I hope) the introduction.
Edit: Changed this comment to a question.
Is on diagram the category of free categories on graphs or is it the same thing as as defined on directed graph?
Thanks Harry. That's what I thought. So and mean the same thing.
We had the definitions of limit and colimit as terminal and initial (co)cones round the wrong way, so have corrected at diagram.
Hi I’m new here. First of all thanks for this great resource you all created. I think there might be a mistake in the component definition of a diagram here. Particularly in the definitions of limiting cone (limit) and limiting co-cone (co-limit). I think it should be stated that the limiting cone is initial among all possible cones and the limiting co-cone is terminal among all possible co-cone. Not the opposite. I might be wrong here, I’m just a hobbyist category theorist ;) Did I misunderstand something ?
No, it’s right as it is, but I think I can understand why some might find it confusing.
Take a simple example, where diagrams are over a discrete category with just two objects. If the diagram consists of objects $X, Y$, then a general cone looks like
$\array{ A \\ \mathllap{f} \downarrow & \searrow \mathrlap{g} & \\ X & & Y }$The limiting cone is the product $X \times Y$ together with its product projections:
$\array{ & & X \times Y \\ & \mathllap{\pi_1} \swarrow & \downarrow \mathrlap{\pi_2} \\ X & & Y }$and for any cone as in the first diagram, there is a unique map of cones to the product cone, given by a map
$A \stackrel{(f, g)}{\to} X \times Y.$Thus the product cone is terminal among all cones: terminal means that for any object there exists a unique map to the terminal.
So we agree, the definition should say it’s terminal, not initial like it is now : over this diagram (def. 2.3) which is universal or initial among all possible cones, in that it ...
<– This is from the definition of a limiting cone in definition 2.4, it should say terminal instead of initial. Same for limiting co-cone.
I see; I was looking at a different part of the page where it was correct. You’re right.
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