2,196 reputation
1139
bio website axiomsofchoice.org
location
age
visits member for 2 years, 5 months
seen Apr 12 at 21:13

A duck walks into a bar. Animal control is promptly called and the duck is released into a near by park.


Apr
9
comment What is the shortest way to write the number $1234567890$?
I computationally searched many numbers of the form $\left|\left(e·a^{b^c+d}\pm f·g^h\right)-1234567890\right|$ and haven't found anything particularly close to zero.
Apr
8
comment Do we lose everything, if the natural transformations in a monad are exactly inverse?
@GiorgioMossa: Thanks for the example. I guess the question doesn't imply that the relation is generally true - I ask if we require the natural transfomrations (which are a family of endomorphisms) are exact inverses/bijections $\eta=\mu^{-1}$, if then the image of $1_{\bf C}$ (the category itself) and the image of $T$ can be distinguished in any way, apart from objects and arrows having different names. If $\eta=\mu^{-1}$, can $T$ add or remove anything to the objects which wasn't there before (like the Maybe monad in Haskell adds a new value to every type.)
Apr
8
comment Do we lose everything, if the natural transformations in a monad are exactly inverse?
@user21929: Ah, okay I see, the index of $\mu$ doesn't coincide with it's domain then. I've updated the questions, hopefully well typed, with the reverse concatenation which should be non-trivial in general. /// As an example I think of $\text{return}(x):=[x]$ and flatten as join (e.g. $\text{join}([[a,b,c],[e],[f,g]]):=[a,b,c,e,f,g]$), which -coming from the list of lists side- can't be inverse to each other. So then if a monad were defined to require the natural morphism components to be strict inverses, would the monad be merely a relabeling or could there still be nontrivial applications?
Apr
8
revised Do we lose everything, if the natural transformations in a monad are exactly inverse?
added 21 characters in body
Apr
8
comment Do we lose everything, if the natural transformations in a monad are exactly inverse?
@user21929: Okay, so does $\mu_{TTA}\circ\eta_{TA}=id_{TA}$ do it? The component of $\eta$ at $TA$ mapps to $TTA$ and the component of $\mu$ at $TTA$ should map back to $TA$ --- Remark: In any case, I should also emphasize that I want to speak of a bijection here (so that the (suitably well typed) reverse order of concatenations of the natural transform is the identity too.)
Apr
8
revised Do we lose everything, if the natural transformations in a monad are exactly inverse?
deleted 78 characters in body
Apr
8
asked Do we lose everything, if the natural transformations in a monad are exactly inverse?
Apr
7
comment Is there a logical interpretation for equalizer and co-equalizer?
Okay, then one last question: I've seen the category of context only in the Bart book and then mentioned in connection with computer science people? Who uses this construct? In particular, I'm a physicist and not convinced it's work looking at it further.
Apr
6
comment How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?
That's what I mean, yes. For some predicate $\phi$ my logic recognises.
Apr
6
revised How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?
deleted 12 characters in body
Apr
6
comment How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?
Thanks for the response. I wonder why not really just say ${\large\frac{{\bf ST}\vdash \phi(a)}{a\in\mathfrak U}}$?
Apr
6
comment Is there a logical interpretation for equalizer and co-equalizer?
You keep saying "contains all free variables", do you emphasize that it might consider more?
Apr
6
accepted Is there a logical interpretation for equalizer and co-equalizer?
Apr
5
comment Is there a logical interpretation for equalizer and co-equalizer?
I think I read about it in Barts "Categorical logic and type theory". Do the formulas essentially represent predicates of properties of the objects of the models? I.e. I.e. an object $\phi$ in $\mathrm{C}(T)$ might be a predicate and a set $M(\phi):=\{x\,|\,\phi(x)\}$ might be the model. Is my interpretation right?
Apr
5
comment If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$
Spend a lot time thinking about this, very tricky. (For one, if $0\in N$ then $f=0$ is a solution.) Consider $f(f(f(x)))$. The function must fulfill $f(c^m·x)=c^m·f(x)$, with $c=3$. And there is a law for iterations of $f$. This means in particular that $f$ must pass any powers of $c$ through and is determined on the numbers $d$ which aren't divisible by $3$. But then, when the double application must multiply the result, how do you store the two data: a) value of input b) number of previous applications. I don't think it works if in- and output is just a number and you make no restrictions.
Apr
5
comment Is there a logical interpretation for equalizer and co-equalizer?
Thanks for the response. I have a little problem to parse the sentence "let {x:t} denote a term in context, which means that t is a term of type A1,...,An;B and x denotes a sequence of variables, including the variables free in t." (Knuth says don't start a sentence with a mathematical symbol ;). So is '{x:t}' or 't' the "term in context"? Do you mean 't' is a term of type 'A1', 'A1,...,An' or 'A1,...,An;B'? I guess the second. As far as I can tell from the sentence below, you consider the category of contexts and arrows are terms. Then M is a functor and the codomain seemingly has products.
Apr
4
comment Advantage of ZF over other set theories such as New Foundation
@Egbert: I'd have some questions, if you're up to it. I'm a physicist and take some formal notes (on the wiki you find on my page). I've written down a set theory axioms for foundations, but I also want to take notes on categories and types (+HoTT). For this I try to patch things together so I get some kind of consistency. In particular I try to squeeze sets into my type system somehow, this is the bulk of it. If you're motivated, please write me a mail (first paragraph) and make some comments/extend what I mean.
Apr
3
comment Is there a logical interpretation for equalizer and co-equalizer?
@Hurkyl: And you say there is no clear logical interpretation, say in terms of higher order logic?
Apr
3
comment Is there a logical interpretation for equalizer and co-equalizer?
Okay. The question arose in thinking about how the subobject classifier of a non-classical topos works (presheaf-category in particular). Since people say this gives a logic and, by the Yoneda lemma, its arrow structure will generally by quite full, don't I get more than just single arrows between two objects?
Apr
3
asked Is there a logical interpretation for equalizer and co-equalizer?