reference request – Explicit homotopy for Hochschild chains from natural isomorphism

Let $A,B$ be $k$-linear (possibly, dg-)categories, let $f,g:Ato B$ be two linear functors, and let $T:fRightarrow g$ be a natural isomorphism.

If one denotes by $C_bullet(A,A)$ the standard Hochschild chain complex of $A$ with coefficients in itself, then $f$ and $g$ induce two chain maps $mathbf{f},mathbf{g}:C_bullet(A,A)to C_bullet(B,B)$.

One knows from general facts about the homotopy theory of dg-categories that there exists a homotopy $mathbf{T}$ between $mathbf{f}$ and $mathbf{g}$.

Here is my question: Is there a reference where one can find an explicit expression for this homotopy?

Ideally, I’d like a reference where the compatibility with the mixed structure on Hochschild chains (that is, with Connes’ boundary operator) is discussed.

One can of course guess (and, with some work, prove) a formula. I’m almost sure such a formula is already written somewhere, but I couldn’t find a reference.

Note that I’d already be happy to know about a reference for (dg-)algebras (in which case a natural isomorphism $T:fRightarrow g$ is just the data of an invertible element $b$ of $B$ such that for every $ain A$, $bf(a)=g(a)b$).

abstract algebra – Second Isomorphism Theorem and Jordan-Holder

In another posting, there was a question about the following:

Let $G$ be a finite non-trivial group with the following two composition series:

${e} = M_0 triangleleft M_1 triangleleft M_2 = G$

${e} = N_0 triangleleft N_1 triangleleft cdots triangleleft N_r = G.$

Prove that $r=2$ and that $G/M_1 cong G/N_1$ and $N_1/N_0 cong M_1/M_0$.

The person posting the question went on to state that “By the second isomorphism theorem I know that $M_1N_2/N_2 cong M_1/(N_2cap M_1)$

Here is my question. In order to use the second isomorphism theorem with $M_1$ and $N_2$ don’t we need to know that $M_1 leq N_G(N_2)$? And if so, then how do we know that $M_1$ actually is in the normalizer of $N_2$ in $G$?

If I can get over this hurdle, I understand the remainder of the original posting. Perhaps this is something obvious, but please help me see whatever it is that I am missing.

Thanks in advance.

linear algebra – Proving that an isomorphism from a vector space $X$ to $X’$ using symmetric bilinear forms is onto

Here’s the problem:

Let $f$ be a symmetric bilinear form over an n-dimensional vector space $X$, and $f$ is non-degenerate (for all nonzero $x in X$, there exists $y in X$ so $f(x,y) neq 0$). Prove that $L_f: X to X’$ defined as $x mapsto f(x, -)$ is an isomorphism.

I can easily prove $L_f$ is one-to-one, and argue that it’s an isomorphism since we are dealing with finite-dimensional spaces. (If $L_f(x) = L_f(y)$ then $f(x-y, z) = 0$ for all $z$. However, since $f$ is non-degenerate, if $x-y neq 0$ then there would exist a $z$ that would make $f$ nonzero. Therefore, $x=y$)

However, I am stuck trying to prove that it is onto, to maybe extend this result to infinite dimensional spaces. Is it even possible?

nt.number theory – Will numerical coincidences remain conjectures forever? Number of groups of order $2^n$ up to isomorphism for $n>3$ is not a palindrome in base 10

Letting $a_n$ be the number of non-isomorphic groups of order $n,$ it appears that $a_n$ is not a palindrome in base $10$ for any $n>3.$ But how do we even begin to prove this? We can use heuristics to argue that if a small palindrome hasn’t been found for sequences growing at least exponentially, then we won’t find one.

But how can we ever convert these heuristics to a proof when there’s no relation between the nature of palindromes in base $10$ and sequences like $a_n$ that aren’t remotely close to having a closed form? It seems like this coincidence and similar such coincidences (consider rings, posets, monoids, and magmas instead) will remain a conjecture forever.

fa.functional analysis – Explicit isomorphism between $L^2(mathbb{R}^2)$ and $L^2(mathbb{R})$?

As Hilbert spaces, $L^2(mathbb{R}^2)$ and $L^2(mathbb{R})$ are isomorphic. Of course the isomoprhism is vastly not unique. I wonder if there are any particularly nice explicit isomorphisms. E.g. I wonder if there is an integral transform
$$
f(x,y) mapsto (K f)(z)=int dx, dy K(x,y,z) f(x,y)
$$

with a nice explicit kernel $K(x,y,z)$ which maps $L^2(mathbb{R}^2)$ isometrically onto $L^2(mathbb{R})$? Any example would be appreciated.

ct.category theory – Word for two morphisms that are equivalent up to right-composition with isomorphism

Let $f:Ato C$, $g:Bto C$ be morphisms in some category.
I call $f,g$ “equivalent” iff there exists an isomorphism $h$ such that $fcirc h=g$ (and consequently $gcirc h^{-1}=f$).

Question: Is there an established term for this kind of equivalence?

Background: In a paper, I am defining this in a slightly more specific setting, and I would like to add a clarifying sentence such as “note that this is the same as the notion of … for general categories”.

vector spaces – Isomorphism between $V^* otimes V$ and End$V$

If $V$ is a vector space and End$V$ is the set of endomorphisms from $V$ to $V$.

Defining a map from $V^*otimes Vrightarrow$ End $V$ by sending some element say, $fotimes v in V^* otimes V$ to endomorphism whose value at $win V$ is $f(w)v$.

I am trying to show this map is an isomorphism between $V^* otimes V$ and End $V$. I am trying to verify it using dual basis.

If some hint can be provided, it will be a great help!