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Oct
30
accepted Where to put the “such that”, given multiple quantifier
Oct
29
comment Where to put the “such that”, given multiple quantifier
Well, it's as unnecessary as using it as a literal translation of "such that" ;) And since it's not part of the official syntax, it all boils down to personal opinion... Anyway, I think you gave a sufficient answer which I might very well accept in the near future.
Oct
29
comment Where to put the “such that”, given multiple quantifier
Yeah, I get that point of view. Although, after finally figuring out what prenexified means, I think I agree more with GitGuts point of view. I.e. seeing the $:$ as a seperator of the string of quantifiers and the quantifier free part in a prenex normal form, instead of a literal "such that".
Oct
29
comment Where to put the “such that”, given multiple quantifier
Yeah, i noticed the absence of $:$ in literature about logic, although the notation seems to be heavily used in papers and lecture notes in different fields. So it seems there's no real right and wrong here...
Oct
29
asked Where to put the “such that”, given multiple quantifier
Oct
20
comment bijective homomorphisms between non isomorphic posets, example , explanation needed
@jhegedus There is no category theoretic isomorphism betwwen $P$ and $Q$. But there is one (non-trivial) from $P$ to $P$, mapping $b$ to $c$ and vice versa.
Oct
7
answered Identifyng objects in a category
Oct
7
comment Trying to understand significance of monoid as a one object category
Considering your update: Define a category $\mathcal{N}$ via: $\operatorname{Ob}(\mathcal{N}) := \{*\}$, $\operatorname{Hom}_\mathcal{N}(*,*) := \{f_i|i\in\mathbb{N}\}$, and composition $f_i\circ f_j = f_{i+j}$. This might seem cheap, but it's easy to see that this $\mathcal{N}$ satisfies every axiom for a category (of course this is only one representant for the categorical monoid, since objects in $\mathcal{Cat}$ are only known up to isomorphisms).
Oct
6
comment Understanding the significance of a functor being full/faithful, especially with adjoints
you can use $\operatorname{Hom}(x,y)\cong\operatorname{Hom}(Rx,Ry)\cong\operatorname{Hom}(LR‌​x,y)$ where the first iso comes from $R$ being fully faithful, and the second from R being right adjoint to $L$.
Sep
30
comment A question regarding Yoneda's lemma.
I'm curious about your example though, seems like you misunderstood some part from the exercise.
Sep
29
comment Prove that the two polynomials intersect each other only at a single point
Ok, i see. I parsed that wrong.
Sep
29
comment Prove that the two polynomials intersect each other only at a single point
"What remains to show is that $D^K_1(\theta)−D^K_2(\theta)$ is neither positive nor negative for all $0<\theta<1/2$ given a $K$" Wait, what? calling the difference $f(\theta)$, and $\theta_0$ the point of intersection, then according to your graphs, $f(\theta)>0$ for $0<\theta<\theta_0$, and $f(\theta)<0$ for $1/2>\theta>\theta_0$
Sep
18
comment 2-category in HoTT: chapter 9 from the HoTT book
Questions should be self-contained, so it'd be nice if you'd actually post the exercise you're refering to.
Sep
10
awarded  Curious
Sep
4
comment Proving a group, $G$, is a group action onto some set, $X$
composition has to hold for arbtrary elements $g_1$, $g_2\in A$. You have only checked the case $g_1 = g_2$ ($=g$, in your proof)
Sep
3
comment Right-adjoint to the inverse image functor
@Patrick Way better than leaving the question unanswered. If you know the answer, you should consider writing one yourself
Sep
2
suggested suggested edit on Let $\displaystyle D= \{z: |x|<1\}$ which of the following is correct
Aug
27
comment Definition of quotient category
There is however a method for computing arbitrary quotients in cat, see e.g. generalized congruences - Epimorphisms in $\mathcal{Cat}$ by Marek A. Bednarczyk et al
Jul
28
comment How do we get a simplicial homology functor?
Beside Zhen Lins embedding into $\mathbf{sSet}$, you could also take the barycentric subdivision of your simplicial complex, which is weakly homotopy equivalent to the original abstract simplicial complex, and has a canonical orientation.
Jul
28
comment are there examples of “category-like” structures where distinct pairs of objects have hom-sets that aren't disjoint?
@bachmanimoff Note, that it's not an unusual POV that the notion of intersections/disjointness only makes sense between subsets. So even though some authors don't mention the disjointness axiom, it doesn't necessarily mean they don't require it. They may just "omit" it because they defined $\operatorname{Hom}(a,b)$ and $\operatorname{Hom}(c,d)$ as different (and hence of course disjoint) sets in the first place.