594 reputation
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bio website natesoares.com
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age 25
visits member for 3 years, 9 months
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Jul
2
awarded  Curious
Dec
18
awarded  Popular Question
Oct
28
accepted What is $R(\omega)$ (and where can I find definitions for similar common notation)?
Oct
28
comment What is $R(\omega)$ (and where can I find definitions for similar common notation)?
Ah, thanks. It helps to know that $R(\omega)$ is called the "rank function".
Oct
28
asked What is $R(\omega)$ (and where can I find definitions for similar common notation)?
Oct
25
accepted What's the motivation behind saturated models?
Oct
24
asked What's the motivation behind saturated models?
Oct
5
accepted Find a set of sentences $\Sigma$ such that the set of all models of $\Sigma$ is countably infinite.
Oct
5
comment Find a set of sentences $\Sigma$ such that the set of all models of $\Sigma$ is countably infinite.
Cool. $s_0 \lor \lnot s_0 \in \Sigma$ and $s_{n^+} \rightarrow (\lnot s_n \land \ldots) \in \Sigma$, $\mathbb M$ contains $\emptyset$ and singleton subsets of $\mathbb S$. Thanks for the hint!
Oct
5
revised Find a set of sentences $\Sigma$ such that the set of all models of $\Sigma$ is countably infinite.
added 4 characters in body
Oct
5
asked Find a set of sentences $\Sigma$ such that the set of all models of $\Sigma$ is countably infinite.
Oct
5
accepted Derive adjoint unit from counit
Oct
5
comment Derive adjoint unit from counit
Thanks. My issue now is that I have trouble proving that $G_{\epsilon_Y} \circ id_{F(G(Y))}^* = id_{G(Y)}$, not because it's particularly difficult, but because I used this fact to prove the naturality of $id_{F(-)}^*$ so I'm not allowed to invoke naturality in my proof. Instead, I invoked the universal mapping property of the counit to claim that $id_{F(-)}^*$ is $\eta$ (and thus is natural), but I don't think this is quite valid. I understand all the parts of the proof, but I haven't had time to sit down and formalize it yet. Hopefully this answer will be sufficient when I do.
Sep
21
asked Derive adjoint unit from counit
Sep
13
accepted Equalizers and Basic limit theorem in Category theory
Sep
13
comment Equalizers and Basic limit theorem in Category theory
Yeah, it seems that $I$, $J$, ... are indeed intended to index the objects $D_I$, $D_J$, ... of $\textbf{D}$. Apologies; this syntax is somewhat new to me.
Sep
13
revised Equalizers and Basic limit theorem in Category theory
added 12 characters in body
Sep
13
comment Equalizers and Basic limit theorem in Category theory
Oh, that was just me being sloppy when summarizing the proof in the book. In the book, $I$ and $J$ are the notation used when referring to elements of $V$ (i.e. only in the indexing of the products), while $D_I$ and $D_J$ are used when referring to objects of $\textbf{D}$ (which are indeed the same, unless I'm missing something). Sorry. I'll go back and edit.
Sep
13
comment Equalizers and Basic limit theorem in Category theory
@KarlKronenfeld, that's precisely what I was missing, thanks. Care to write up an answer?
Sep
13
comment Equalizers and Basic limit theorem in Category theory
But limits require that for any limit $X$ with arrow $f_i : X \rightarrow D_i$ in the limit and arrow $g : D_i \rightarrow D_j$ in the diagram, $f_j = g \circ f_i$. If we ignore extra $D_e : D_I \rightarrow D_J$ in the proof then it seems like there are arrows in the diagram (the ignored arrows) that could violate this equation.