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Presently I am concerned with the following: I want to teach some basics of limits and colimits of topological spaces to undergraduates. I had tried to gently introduce some category theoretic terminology as I introduced topological spaces as such, but a little testing reveals that part of the audience would rather not see these side remarks turn out to become required reading.
But now since all the shapes of diagrams that I’d be about to consider anyway are free, I figured I circumvent the need to speak of diagrams as functors by restricting to free domains and simply giving everything explicitly in components, with the underlying category theory again relegated to side-remarks that may be ignored at will.
I am trying my hands at an exposition of this kind in a new entry free diagram.
Eventually this kind of material might also be worthwhile as introductory exposition at limits and colimits by example.
In fact at commuting diagram all pointers to “diagram” are meant to go to “free diagram”, so I changed them. I also replaced all occurences of “quiver” there with “free category”.
A free diagram, as it is currently defined, is the same thing as a morphism of directed graphs. Perhaps this is worthy of pointing out. (And a category is a monoid in directed graphs, so a free category is the same thing as a free monoid.)
A free diagram, as it is currently defined, is the same thing as a morphism of directed graphs.
Maybe not quite. While functors out of the free category on a directed graph $D$ into some category $\mathcal{C}$ (which is what the entry considers) are in natural bijection to homorphisms from $D$ to the underlying graph of $\mathcal{C}$, the latter are strictly speaking not themselves diagrams in $\mathcal{C}$, as they are not functors to $\mathcal{C}$.
I should say that I am using this kind of exposition in section 6 of Introduction to Topology – 1.
(That section and the following ones are still under construction. Sections 1-5 though are pretty much polished.)
the latter are strictly speaking not themselves diagrams in $\mathcal{C}$, as they are not functors to $\mathcal{C}$.
Except as far as I can tell, your definition of free diagram is not phrased in terms of functors – it really is the same as a morphism of directed graphs.
Okay, I have expanded the entry, hopefully now it’s clearer.
I re-iterate that the motivation for this entry was not the desire to record the abstract definition of free diagrams, which is trivial, but to use this special case to give an expository discussion of limits and colimits that entirely avoids the language of categories and functors, and just expresses everything in components.
Earlier at diagram – Component definition I had tried this kind of exposition in the general case, for possibly non-free shapes. But (obvious as that may have been from the outset) this didn’t really give a low-weight exposition as intended. So then I came to think that admitting for non-free shape categories is overkill for an exposition anyway, since the bulk of first examples of limits/colimits that one will want to display in an exposition are all over just free diagrams.
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