Related papers: The List Object Endofunctor is Polynomial

Determining the range of complex maps plays a fundamental role in the study of several complex variables and operator theory. In particular, one is often interested in determining when a given holomorphic function is a self-map of the unit…

We apply the notion of relative adjoint functor to generalise closed monoidal categories. We define representations in such categories and give their relation with left actions of monoids. The translation of these representations under lax…

We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system),…

Let $G$ be a topological commutative semigroup with unit. We prove that a continuous function $f\colon G\to \cc$ is a generalized exponential polynomial if and only if there is an $n\ge 2$ such that $f(x_1 +\ldots +x_n )$ is decomposable;…

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful;…

Let $V$ be a vector space over a field $k, P:V\to k, d\geq 3$. We show the existence of a function $C(r,d)$ such that $rank (P)\leq C(r,d)$ for any field $k,char (k)>d$, a finite-dimensional $k$-vector space $V$ and a polynomial $P:V\to k$…

We study generalizations of Schur functors from categories consisting of flags of vector spaces. We give different descriptions of the category of such functors in terms of representations of certain combinatorial categories and infinite…

Given symmetric monoidal infinity-categories C and D, subject to mild hypotheses on D, we define an infinity-categorical analog of the Day convolution symmetric monoidal structure on the functor category Fun(C, D). An E_infinity monoid for…

We study rank functions on a triangulated category $\mathcal{C}$ via its abelianisation $\operatorname{mod}\mathcal{C}$. We prove that every rank function on $\mathcal{C}$ can be interpreted as an additive function on…

The notion of an $\mathcal{M}$-coextensive object is introduced in an arbitrary category $\mathbb{C}$, where $\mathcal{M}$ is a distinguished class of morphisms from $\mathbb{C}$. This notion allows for a categorical treatment of the strict…

In this paper we investigate a categorical aspect of $n$-trivial extension of a ring by a family of modules. Namely, we introduce the right (resp., left) $n$-trivial extension of a category by a family of endofunctors. Among other results,…

Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if $g(h(x))$ has boundedly many terms, then $h(x)\in \C[x]$ must also have boundedly many terms. Solving an older conjecture raised by R\'enyi and by…

For an endofunctor $H$ on a hyper-extensive category preserving countable coproducts we describe the free corecursive algebra on $Y$ as the coproduct of the final coalgebra for $H$ and the free $H$-algebra on $Y$. As a consequence, we…

We discuss here a generalization of a theorem by Dunfield stating that the peripheral holonomy map, from the character variety of a 3-manifold to the A-polynomial is birational. Dunfield's proof involves the rigidity of maximal volume. The…

We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets).…

Suppose $(\mathcal{C},\mathbb{E},\mathfrak{s})$ is an $n$-exangulated category. We show that the idempotent completion and the weak idempotent completion of $\mathcal{C}$ are again $n$-exangulated categories. Furthermore, we also show that…

Let $\mathcal{B}$ be a subcategory of a given category $\mathcal{D}$. Let $\mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $\mathcal{B}$ to $\mathcal{D}$ such that $\mathcal{B}$…

Given a simplicial idempotent augmented endofunctor $F$ on a simplicial combinatorial model category $M$, under the assumption of Vopenka's principle, we exhibit a set $A$ of cofibrant objects in $M$ such that $F$ is equivalent to $\CW_A$,…

If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional…

If T is a commutative monad on a cartesian closed category, then there exists a natural T-bilinear pairing from T(X) times the space of T(1)-valued functions on X ("integration"), as well as a natural T-bilinear action on T(X) by the space…