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Added a diagram to cone and changed some notation to be compatible with cone morphism and Understanding Constructions in Set
Could someone help me unwrap the definition of cone? I don't understand the leap where it goes from natural transformation to components.
I'm sure it is just the definition of natural transformation, which I haven't internalized yet. Anyway, I created ericforgy:Cone. Feedback welcome.
Edit: I see. Finn already walked me through it at cone morphism, but I was too dense to realize. I'll get the hang of this stuff!
BTW, you can create an easy link to http://ncatlab.org/ericforgy/show/Cone as [[ericforgy:Cone]]
: Cone (ericforgy).
Or [[ericforgy:Cone|ericforgy:Cone]]
if you want it to be clear that its from your web: ericforgy:Cone. (Maybe we should convice Andrew to do that automatically, as Instiki itself does.)
At cone the subsection As a contracted cylinder is not about the category-theoretic notion of cone but the homotopy-theoretic notion (in fact the same diagram is reproduced at mapping cone). There is the notion of the cone of a simplicial set, explained at join of simplicial sets, which seems to be the same thing; a few pages on simplicial/(oo,1)-categorical stuff (see the Linked from bit at the end of cone, or e.g. filtered (infinity,1)-category, where I first noticed this) link to cone for this notion, though it isn’t explained there. Should there be a separate page on cones in topology/homotopy theory?
At the heart of it there is the same concept at play in all cases, which induces a notion of cone from an interval object / cylinder object. Cones over diagram categories (or their images in some other category) are in this sense on the same footing as cones in topology.
But I certainly agree that this could be made clearer in the entiries. If you (or somebody else) decides to split off entries, I’d suggest having the main entry give the general idea and then pointing to the special cases.
Just to jot something down quickly for Eric: given any diagram scheme $J$, you can create a new scheme $J_+$ by formally adjoining an initial object. (If $J$ already has an initial object, you add a new one anyway; the old one will not be initial in $J_+$. Then, a cone over $J$ is really the same thing as a diagram $J_+ \to C$. Dually, a cocone over $J$ is a diagram $J^+ \to C$, where $J^+$ is obtained by adjoining a terminal object.
This construction $J_+$ fits well the meaning of cone in topology, or of general cylinder objects.
Oh, OK, I think I get the idea – just as the simplicial/homotopical cone is the homotopy pushout of the identity along the unique map to the terminal object, the categorical cone should be the cocomma object of those two. Then a cone over an I-diagram in C will be the same thing as a functor $cone(I) \to C$. So probably cone should explain both notions. I may not get around to fixing up the entry for a while, though.
[Edit: crossed with Todd’s reply; I think we’re talking about the same thing.]
I’ve made some edits along the above lines to cone. See what you think.
Thanks! Nice.
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