@JulianRachman I'm sorry, I don't think I understand the stuff about decreasing sequences. I mean, certainly if the category $C$ is a set with a partial order or something then your definition seems okay (modulo the fact about it becoming constant), but I don't think one can make any such definition for an arbitrary category.
You will have time to learn how to write introductions.
I am concerned though about your definition of $\sqsubseteq$ and ``infinitely decreasing sequences" in part 2.1 however.
I mean, given an arbitrary set $A$ and a sequence $\{a_i\}$ in $A$, it literally means nothing to say $a\sqsubseteq b$ unless you specify that $A$ is a set with a preorder (or some kind of order) denoted $\sqsubseteq$ or $\sqsubset$.
For instance, if I talk about an infinite decreasing sequence in $\mathbb{R}$, denoted $\{a_i\}$, this means that $a_{i+1}\leq a_i$ for all $i$.
But note, I'm using the fact that $\mathbb{R}$ already has a notion of less than or greater than attached to it.
Many have said that they have not heard of Higman's Lemma before. However I do not want that to be what I start with in my introduction like "No one knows about Higman's Lemma. However in this paper,..."
Ya. I have been. I take a look at Lurie's every once and a while. However his introductions are filled with just "Let "this" be "this." Then 'this' is "'this' iff 'this' is 'that.'"
I am actually currently reading all of the introductions of every paper regarding Higman's Lemma and wqo's with bright highlighting.
And would you mind taking a look at my abstract and see if you were a mathematician looking for something to read if you would skim may paper because of my abstract?
Unfortunately I don't think that I would skim the paper, primarily because I don't really know anything about, say, order theory, and have never heard of Higman's Lemma. So I'm not sure I can really say whether or not the abstract would pique my interest.
I do think it could use to be more concise however. Something like "We give a new proof of Higman's Lemma as well as a number of new applications. These results are obtained by suitable categorification of the Lemma, using categories of preorders, monoids and preordered monoids. Our proof allows us to avoid unnecessary reference to computability." Or something like this. I dunno. If you've got people reading your stuff and mentoring you, listen to them.
Anyway, I've got to go to bed. But good luck. You're certainly much further along than I was in high school (at which time I hated mathematics and thought I was going to be a either a poet, a vagabond, or both).
@QiaochuYuan I like Sublime Text + LaTeXing, but both are commercial products. My criteria are that it should have functional autocomplete for \ref and \cite for multi-file projects. It was surprising how few editors do it right.
I'm with @ZhenLin on this one. Sublime Text is really great. The only thing I don't know how to make it do is to trick it into doing things like typing \infty when I type the keystroke Option+5 to get ∞ (which happens automatically in TexShop).
Anyway, if you want to set up your own key bindings, you can: add {"keys": ["Option+5"], "command": "insert", "args": {"characters": "\\infty"}} to the LaTeXing key bindings file.
@QiaochuYuan taking full advantage of the functionalities offered by TeXstudio may take time, but allowing this to happen is a rewarding decision
my favourite features are: character level sync, thumbnails for references and citations, cmd + click on a reference travels to the label, and live partial compilations
