ct.category theory – Additivization of functors in an abelian monoidal category (crosspost from MSE)

I posted a question a week ago on math.stackexchange. As is sometimes the case, I got no answers. Considering that the question is about a research article, I hope that it might be relevant for MathOverflow.

Here is the original question:


I’m having trouble with the proof of Lemma 2.9 in “Cohomology of Monoids in Monoidal Categories” by Baues, Jibladze, and Tonks, and I was wondering if someone could clarify a detail. I’ll try to summarize the context of the lemma.

Context

Let $(Bbb A,circ,I)$ be an monoidal category where $Bbb A$ is abelian: in particular, $circ$ is not necessarily additive in both arguments. Suppose that $circ$ is left distributive, i.e. the natural transformation
$$(X_1circ Y)oplus(X_2circ Y)rightarrow (X_1oplus X_2)circ Y$$
is an isomorphism. For example, $Bbb A$ could be the category of linear operads (this is a motivating example of the article). Given an endofunctor $F$ of $Bbb A$, we define its cross-effect
$$F(A|B):=ker(F(Aoplus B)rightarrow F(A)oplus F(B)).$$
The additivization of $F$ is then the functor $F^text{add}$ defined by
$$F^text{add}(A):=text{coker}left(F(A|A)rightarrow F(Aoplus A)xrightarrow{F(+)}F(A)right).$$
The idea is that $F^text{add}$ is the additive part of $F$.

Let $(M,mu,eta)$ be an internal monoid in $Bbb A$, and let $L_0$ be the endofunctor of $Bbb A$ defined by $L_0(A)=Mcirc(Moplus A)$. Let $L:=L_0^text{add}$ be the additivization of $L_0$. (In the case of operads, represented as planar trees, I see $L(A)$ as the space of trees whose nodes are all labeled by elements of $M$ except for one leaf, which is labeled by an element of $A$.)

Suppose now that $Bbb A$ is right compatible with cokernels, i.e. that

for each $AinBbb A$, the additive functor $Acirc-:Bbb ArightarrowBbb A$ given by $Bmapsto Acirc B$ preserves cokernels.

Then, in the proof of Lemma 2.9, the authors claim the following:

By the assumption that $Bbb A$ is right compatible with cokernels it follows that $L(L(X))$ is the additivisation of $L_0(L_0(X))$ in $X$ (…).

Remarks

If anyone could provide an explanation of the last claim, I would be very grateful. However, my inability to understand how to show this might be related to two other issues I have:

1) Elsewhere in the literature, cross-effects are only defined when $F$ is reduced, i.e. $F(0)=0$ (e.g. here, section 2). But we can always reduce a functor by taking the cokernel of $F(0)rightarrow F(X)$, so I don’t think it’s much of a problem.

2) In the first quote, the authors state that $Acirc -$ is additive, which is quite the opposite of the initial hypothesis that $circ$ be left distributive, and not necessarily right distributive. How to resolve this apparent conflict?

nt.number theory – On the $mathsf{LCM}$ of a set of integers defined by moduli of powers

For integers $a,b,t$ define $$mathcal R_t(a,b)={qinmathbb Zcap(1,min(a^t,b^t)): a^tequiv b^tbmod q}$$ and $mathsf{LCM}(mathcal R_t(a,b))$ to be $mathsf{LCM}$ of all entries in $mathcal R_t(a,b)$.

Similar reasoning to On $mathsf{LCM}$ of a set of integers gives $$mathsf{LCM}(mathcal R_t(a,b))leqmathsf{LCM}(T_t(a,b))$$ where $T_t(a,b)$ is defined as $$T_t(a,b)=Big{qinmathbb Zcap(1,infty):q|Big((a-b)sum_{i=0}^{t-1}a^{t-1-i}b^iBig)Big}.$$

So $mathsf{LCM}(mathcal R_t(a,b))=mathsf{LCM}(T_t(a,b))$ holds.

If $a,binbig(frac r2,rbig)$ hold and are coprime then what is the probability $mathsf{LCM}(mathcal R_t(a,b))<beta r^{t-alpha}$ at some $alphain(0,t)$ and $beta>0$?

set theory – Why are quotient sets (types) called quotients — are they the inverse of some product?

There seems to be a beautiful relation between natural numbers and sets (and types),
as in the size of a discriminate union, cartesian product, and function type,
is described by the sum, product, exponential of the sizes of the components. (As I learned from type theory). This also makes it easy to see why the symbols + and x are used for discriminate union and cartesian product (sum type and product type).

$$
forall A, B, C : text{sets} \
A + B = C ~ implies ~ |A| + |B| = |C|\
A times B = C ~ implies ~ |A| times |B| = |C|\
A → B = C ~ implies ~~~~~~~~ |B|^{|A|} = |C|
$$

However, why are quotient sets (and quotient types) called quotients and use the symbol $/$?

That does not seem to make sense to me. At the very least, to deserve the name quotient, I would expect them to somehow be the inverse of some product. I first thought they should be the inverse of the cartesian product, I tried to google this, but I cannot find anything. Is there some relation between quotient (sets) and (cartesian) products, that I am missing?

nt.number theory – Reciprocity theorem with n >= 5

My equation is:

$2^{(p-1)/n} equiv 1 pmod p$

And this is always true: $1 equiv p pmod n$

I want to know which form $p$ must have to make equation 1 true.

I know for $n=3$ (cubic reciprocity) the form is $p=x^2+27y^2$.

For $n=4$ (biquadratic reciprocity) the form is $p=x^2+64y^2$

I know the solution has to deal with Gaussian integers and Artin’s reciprocity theorem.

Can someone please show me how to exactly do this maybe with the example $n=5$.

number theory – On quadratic form $L^2+27M^2$

If $4q=L^2+27M^2$, where $qequiv 1 (mod 3)$ is a power of $p$. How can I show that there is only one pair of integers (L,M) satisfying the condition where $Lequiv 1 (mod 3)$ and M up to sign.

I have proved by elementary number theory that there is at most one pair of POSITIVE integers. But what if $4q=L’^2=L^2+27M^2$, where $Lequiv L’equiv 1$?

measure theory – Prove that the composition of a measurable function with a continuous function results into a measurable function

Let $Omega$ be a measurable subset of $Bbb R^n$, and let $W$ be an open subset of $Bbb R^m$. If $f:Omegato W$ is measurable, and $g:WtoBbb R^p$ is continuous, then $gcirc f:OmegatoBbb R^p$ is measurable.

MY ATTEMPT

Let $VsubseteqBbb R^p$ be an open set. Then we have that $g^{-1}(V)$ is open. Since $f$ is measurable, $f^{-1}(g^{-1}(V))$ is also measurable.

In other words, for every open set $VsubseteqBbb R^p$, the set $(gcirc f)^{-1}(V)$ is measurable, and the result holds.

I am a little bit unsure about the wording of the proof. Could someone please check my arguments?

fa.functional analysis – A question about Schwartz-type functions used in analytic number theory

In analytic number theory we like to weigh our counting functions with a smooth function $f$, so that we may apply Poisson’s summation formula and take advantage of Fourier transforms. Typically the weight function $f$ will be a Schwartz type function with the following properties:

1) $f(x) geq 0 $ for all $x in mathbb{R}$;

2) $f(x) = 1$ for $x in (-X,X)$ say;

3) $f(x) = 0$ for $|x| > X + Y$; and

4) $f^{(j)}(x) ll_j Y^{-j}$ for $j geq 0$.

In most applications the dependence on $j$ in the last condition does not matter, since $j$ would be bounded. However, in a problem I am considering it might be worthwhile to make $j$ a (slow growing) function of $X$ so it then becomes relevant to know how the bound depends on $j$. Is it possible to give an explicit example of a function $f$ satisfying the above properties for which the dependence can be made explicit?

nt.number theory – Number of distinct sum of two and four squares under sum condition

Given $0leq mleq n$ and $x_1+dots+x_4=n$ holds what is the

  1. distribution of

  2. maximum

number of distinct values of the sum of squares function $$x_1^2+dots+x_4^2$$ on conditions

a. $0=x_1=x_2leq x_3,x_4leq m$

b. $0leq x_1,x_2,x_3,x_4leq m$ holds?

(Note we do not force order).

complexity theory – Dividing students into 4 groups based on preferences is NP-complete

Given a set of students $H$ of size $n$, and a set $E subseteq H times H $ of pairs of students that dislike each other, we want to determine whether it’s possible to divide them into $4$ groups such that:

  • no two students that dislike each other end up in the same group,
  • the size of each group must be at least $frac{n}{5}$.

I want to prove that this problem is NP-complete. I suspect that I could use the NP-completeness of the independence set problem, yet I have some problems with finding an appropriate reduction.

Let $G = (H, E)$ an undirected graph – each edge represents two students that dislike each other.

For the groups to be of the required size, their size must be $k in left (frac{n}{5}, frac{2n}{5} right ) cap mathbb{N}$. I could then try checking whether there is an independence set of size $k$ (which would mean there are $k$ students that potentially like each other), remove its vertices, and repeat for the next $k$. However, I don’t think this would result in a polynomial number of size combinations.

Do you have any advice on constructing this reduction?

group theory – Elements of $mathbb{F}_7^*/mathbb{F}_7^{*3}$

I think I have forgotten some basic group theory, but I am having hard time representing the elements from $mathbb{F}_7^*/mathbb{F}_7^{*3}$, where $mathbb{F}_7^{*3}$ denotes all elements that are cubes in $mathbb{F}_7^*$. I have figured out that $mathbb{F}_7^{*3} = {bar{1},bar{6}}$ and hence $mathbb{F}_7^*/mathbb{F}_7^{*3}$ is isomorphic to $mathbb{Z}/3mathbb{Z}$. However, I am looking for representative elements from $mathbb{F}_7^*$. Any help would be appreciated.