This question is not really about elementary topoi, it is much more about a category $(\mathcal{E}, \Omega)$ admitting a subobject classifier, or about a category with power objects, you can choose the context that inspires you the most. Of course, elementary topoi are the go-to example. Q: Can ...

Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's classification of orthogonal polynomials of A-type 0 into what are now known as Laguerre, Hermite, Charl...

I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $X\sim \mathrm{GUE} (d)$, where $\mathrm{GUE} (d)$ stands for Gaussian Unitary Ensemble in dimension $d$. The expression for...

Let $k$ be an infinite perfect field in positive characteristic $p$, i.e. every element of $k$ is a $p$th power. I am interested in properties of finite fields that can be extended to $k$. For example: Let $L$ be a finite Galois extension of $k$. Is $\mathrm{Gal}(L/k)$ always cyclic? Let $L$ and...

Let $A$ be a non-degenerate algebra and let $\Delta: A \to M(A \otimes A)$ be a non-degenerate morphism. We can extend the algebra morphism $$\iota \otimes \Delta: M(A \otimes A) \to M(A \otimes A \otimes A)$$ Suppose I want to show that ($1$ is the unit of $M(A))$: $$(\iota \otimes \Delta)(x \ot...

Consider the following fragment from the paper "Multiplier Hopf-algebras" by Van Daele. Can someone explain how the coassociativity in definition 2.2 (ii) and the requirement $(\Delta \otimes \iota)\Delta = (\iota \otimes \Delta)\Delta$ are equivalent? I understand how we can extend these morphi...