Related papers: Products in a Category with Only One Object
In an additive factorial monoid each element can be represented as a linear combination of irreducible elements (atoms) with uniquely determined coefficients running over all natural numbers. In this paper we develop for a wide class of…
Graph products of monoids provide a common framework for direct and free products, and graph monoids (also known as free partially commutative monoids). If the monoids in question are groups, then any graph product is, of course, a group.…
For a given twisted cartesian products of simplicial sets, we construct the corresponding twisted tensor product in the sense of Brown, with an explicit twisting function whose formula is simple without using inductions. This is done by…
The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups)in an ambient group $G$. Their work gives…
We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all $n \times n$ upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative…
We apply the free product construction to various local algebras in algebraic quantum field theory. If we take the free product of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism, we obtain a…
We present a simple extension of the classical Hilton-Eckmann argument classically used to prove that the endomorphism monoid of the unit object in a monoidal category is commutative. It allows us to recover in a uniform way well-known…
We provide a categorical framework for mathematical objects for which there is both a sort of "independent" and "dependent" composition. Namely we model them as duoidal categories in which both monoidal structures share a unit and the first…
In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. These are…
In Fritz & Rischel, Infinite products and zero-one laws in categorical probability, the problem was posed of finding an interesting Markov category which is causal and has all (small) Kolmogorov products (there Problem 6.7). Here we give an…
A concrete model of the free skew-monoidal category Fsk on a single generating object is obtained. The situation is clubbable in the sense of G.M. Kelly, so this allows a description of the free skew-monoidal category on any category. As…
Categorical models of the exponential modality of linear logic will often, but not always, support an operation of differentiation. When they do, we speak of a monoidal differential modality; when they do not, we have merely a monoidal…
Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major…
We study automaton structures, i.e. groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we…
In this paper, we present an infinity-categorical version of the theory of monoidal categories. We show that the infinity category of spectra admits an essentially unique monoidal structure (such that the tensor product preserves colimits…
We consider Turing machines as actions over configurations in $\Sigma^{\mathbb{Z}^d}$ which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines…
From the analogue of Boehm's Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian…
The word problem for categories with free products and coproducts (sums), SP-categories, is directly related to the problem of determining the equivalence of certain processes. Indeed, the maps in these categories may be directly…
Let $D$ be a division ring of fractions of a crossed product $F[G,\eta,\alpha]$ where $F$ is a skew field and $G$ is a group with Conradian left-order $\leq$. For $D$ we introduce the notion of freeness with respect to $\leq$ and show that…
In this note, we first work out some `bare hands' computations of the most elementary possible free products involving $\mathbb{C}^2 ~(=\mathbb{C} \oplus \mathbb{C} $) and $M_2 ~(= M_2(\mathbb{C}))$. Using these, we identify all free…