Levels in the toposes of simplicial sets and cubical sets

The essential subtoposes of a fixed topos form a complete lattice, which gives rise to the notion of a level in a topos. In the familiar example of simplicial sets, levels coincide with dimensions and give rise to the usual notions of n-skeletal and n-coskeletal simplicial sets. In addition to the obvious ordering, the levels provide a stricter means of comparing the complexity of objects, which is determined by the answer to the following question posed by Bill Lawvere: when does n-skeletal imply k-coskeletal? This paper answers this question for several toposes of interest to homotopy theory and higher category theory: simplicial sets, cubical sets, and reflexive globular sets. For the latter, n-skeletal implies (n+1)-coskeletal but for the other two examples the situation is considerably more complicated: n-skeletal implies (2n-1)-coskeletal for simplicial sets and 2n-coskeletal for cubical sets, but nothing stronger. In a discussion of further applications, we prove that n-skeletal cyclic sets are necessarily (2n+1)-coskeletal.Comment: This paper subsumes earlier work of the first, third, and fourth authors. 19 page

Linearization of graphic toposes via Coxeter groups

AbstractIn an associative algebra over a field K of characteristic not 2, those idempotent elements a, for which the inner derivation [−,a] is also idempotent, form a monoid M satisfying the graphic identity aba=ab. In case K has three elements and M is such a graphic monoid, then the category of K-vector spaces in the topos of M-sets is a full exact subcategory of the vector spaces in the Boolean topos of G-sets, where G is a crystallographic Coxeter group which measures equality of levels in the category of M-sets

The algorithmics of solitaire-like games

One-person solitaire-like games are explored with a view to using them in teaching algorithmic problem solving. The key to understanding solutions to such games is the identification of invariant properties of polynomial arithmetic. We demonstrate this via three case studies: solitaire itself, tiling problems and a novel class of one-person games. The known classification of states of the game of (peg) solitaire into 16 equivalence classes is used to introduce the relevance of polynomial arithmetic. Then we give a novel algebraic formulation of the solution to a class of tiling problems. Finally, we introduce an infinite class of challenging one-person games, which we call ``replacement-set games'', inspired by earlier work by Chen and Backhouse on the relation between cyclotomic polynomials and generalisations of the seven-trees-in-one type isomorphism. We present an algorithm to solve arbitrary instances of replacement-set games and we show various ways of constructing infinite (solvable) classes of replacement-set games

On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos

In the effective topos there exists a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for chain-complete lattices do not have constructive (topos-valid) proofs

The Hopf algebra of Möbius intervals

An unpublished result by the first author states that there exists a Hopf algebra H such that for any Moebius category C (in the sense of Leroux) there exists a canonical algebra morphism from the dual H* of H to the incidence algebra of C. Moreover, the Moebius inversion principle in incidence algebras follows from a `master´ inversion result in H*. The underlying module of H was originally defined as the free module on the set of iso classes of Moebius intervals, i.e. Moebius categories with initial and terminal objects. Here we consider a category of Moebius intervals and construct the Hopf algebra via the objective approach applied to a monoidal extensive category of combinatorial objects, with the values in appropriate rings being abstracted from combinatorial functors on the objects. The explicit consideration of a category of Moebius intervals leads also to two new characterizations of Moebius categories.Fil: Lawvere, F. W.. No especifíca;Fil: Menni, Matías. Ministerio de Educación, Cultura, Ciencia y Tecnología. Secretaria de Gobierno de Ciencia Tecnología e Innovación Productiva. Agencia Nacional de Promoción Científica y Tecnológica. Fondo Argentino Sectorial; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Characterization of quantale-valued metric spaces and quantale-valued partial metric spaces by convergence

[EN] We identify two categories of quantale-valued convergence tower spaces that are isomorphic to the categories of quantale-valued metric spaces and quantale-valued partial metric spaces, respectively. As an application we state a quantale-valued metrization theorem for quantale-valued convergence tower groups.Jäger, G.; Ahsanullah, TMG. (2018). Characterization of quantale-valued metric spaces and quantale-valued partial metric spaces by convergence. Applied General Topology. 19(1):129-144. doi:10.4995/agt.2018.7849SWORD12914419

The Hopf algebra of Möbius intervals

An unpublished result by the first author states that there exists a Hopf algebra H such that for any Moebius category C (in the sense of Leroux) there exists a canonical algebra morphism from the dual H* of H to the incidence algebra of C. Moreover, the Moebius inversion principle in incidence algebras follows from a `master´ inversion result in H*. The underlying module of H was originally defined as the free module on the set of iso classes of Moebius intervals, i.e. Moebius categories with initial and terminal objects. Here we consider a category of Moebius intervals and construct the Hopf algebra via the objective approach applied to a monoidal extensive category of combinatorial objects, with the values in appropriate rings being abstracted from combinatorial functors on the objects. The explicit consideration of a category of Moebius intervals leads also to two new characterizations of Moebius categories.Fil: Lawvere, F. W.. No especifíca;Fil: Menni, Matías. Ministerio de Educación, Cultura, Ciencia y Tecnología. Secretaria de Gobierno de Ciencia Tecnología e Innovación Productiva. Agencia Nacional de Promoción Científica y Tecnológica. Fondo Argentino Sectorial; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Homotopy nilpotent groups

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $\pi_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $\Omega F$ have values in homotopy n-nilpotent groups.Comment: 16 pages, uses xy-pic, improved exposition, submitte

The Biequivalence of Locally Cartesian Closed Categories and Martin-L\"of Type Theories

Seely's paper "Locally cartesian closed categories and type theory" contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-L\"of type theories with Pi-types, Sigma-types and extensional identity types. However, Seely's proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seely's theorem: that the B\'enabou-Hofmann interpretation of Martin-L\"of type theory in locally cartesian closed categories yields a biequivalence of 2-categories. To facilitate the technical development we employ categories with families as a substitute for syntactic Martin-L\"of type theories. As a second result we prove that if we remove Pi-types the resulting categories with families are biequivalent to left exact categories.Comment: TLCA 2011 - 10th Typed Lambda Calculi and Applications, Novi Sad : Serbia (2011

Categorical Foundation of Quantum Mechanics and String Theory

The unification of Quantum Mechanics and General Relativity remains the primary goal of Theoretical Physics, with string theory appearing as the only plausible unifying scheme. In the present work, in a search of the conceptual foundations of string theory, we analyze the relational logic developed by C. S. Peirce in the late nineteenth century. The Peircean logic has the mathematical structure of a category with the relation $R_{ij}$ among two individual terms $S_i$ and $S_j$, serving as an arrow (or morphism). We introduce a realization of the corresponding categorical algebra of compositions, which naturally gives rise to the fundamental quantum laws, thus indicating category theory as the foundation of Quantum Mechanics. The same relational algebra generates a number of group structures, among them $W_{\infty}$. The group $W_{\infty}$ is embodied and realized by the matrix models, themselves closely linked with string theory. It is suggested that relational logic and in general category theory may provide a new paradigm, within which to develop modern physical theories.Comment: To appear in International Journal of Modern Physics