Related papers: Associahedra, Multiplihedra and units in $A_{\inft…
We introduce a minimal ZFC-internal axiom system for pre-structural data (X, A, mu, mu^{otimes 2}, R, I, Pi_R, G, E_0, eta), where Pi_R : X -> R is a designated map and G subset X x X is a measurable relation; admissible structural models…
We verify a confluence result for the rewriting calculus of the linear category introduced in our previous paper. Together with the termination result proved therein, the generalized coherence theorem for linear category is established.…
We prove an inhomogeneous analogue of W. M. Schmidt's (1969) theorem on Hausdorff dimension of the set of badly approximable systems of linear forms. The proof is based on ideas and methods from the theory of dynamical systems, in…
To characterize categorical constraints - associativity, commutativity and monoidality - in the context of quasimonoidal categories, from a cohomological point of view, we define the notion of a parity (quasi)complex. Applied to groups…
Higher derivations on an associative algebra generalizes higher order derivatives. We call a tuple consisting of an algebra and a higher derivation on it by an AssHDer pair. We define a cohomology for AssHDer pairs with coefficients in a…
For a set $E \subseteq \mathbb{F}_q^d$, the distance set is defined as $\Delta(E) := \{\|\mathbf{x} - \mathbf{y}\| : \mathbf{x}, \mathbf{y} \in E\}$, where $\|\cdot\|$ denotes the standard quadratic form. We investigate the…
The logic of definitions is a family of logics for encoding and reasoning about judgments, which are atomic predicates specified by inference rules. A definition associates an atomic predicate with a logical formula, which may itself depend…
The tree-level scattering amplitudes for $\text{tr}(\phi^3)$ theory can be interpreted as a sum over the vertices of a polytope known as the associahedron. For each graph $G$, there exists a natural generalisation of the associahedron,…
Quantum coherence serves as a crucial physical resource, with its quantification emerging as a focal point in contemporary research. Superadditivity constitutes one of the most fundamental attributes in characterizing the coherence…
This paper is the second in a series of two papers about generalizing Quillen's Theorem A to strict $\infty$-categories. In the first one, we presented a proof of this Theorem A of a simplicial nature, direct but somewhat ad hoc. In the…
On metric spaces equipped with doubling measures, we prove that a differentiability theorem holds for Lipschitz functions if and only if the space supports nontrivial (metric) derivations in the sense of Weaver that satisfy an additional…
Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian…
The logic underlying the Abella proof assistant includes mechanisms for interpreting atomic predicates through fixed point definitions that can additionally be treated inductively or co-inductively. However, the original formulation of the…
We identify a strong structural obstruction to Uniform Separation in constructive arithmetic. The mechanism is independent of semantic content; it emerges whenever two distinct evaluator predicates are sustained in parallel and inference…
The (A)CGW categories of Campbell and Zakharevich show how finite sets and varieties behave like the objects of an exact category for the purpose of algebraic $K$-theory. These structures admit a well-behaved Q-construction akin to…
A new construction to associate an internal category to an enriched one is presented. The key concept is that of extensive ambient category, and the construction follows the one that associates a category whose idempotents split to a given…
We show that in a weak globular $\omega$-category, all composition operations are equivalent and commutative for cells with sufficiently degenerate boundary, which can be considered a higher-dimensional generalisation of the Eckmann-Hilton…
We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several examples, including a homotopy theory for…
Induction is the process by which we obtain predictive laws or theories or models of the world. We consider the structural aspect of induction. We answer the question as to whether we can find a finite and minmalistic set of operations on…
In \cite{Ca2016} and \cite{Ca2018}, we introduced a notion of effective reducibility between set-theoretical $\Pi_{2}$-statements; in \cite{Ca2025}, this was extended to statements of arbitrary (potentially even infinite) quantifier…