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We give an introduction to logic tailored for algebraists, explaining how proofs in linear logic can be viewed as algorithms for constructing morphisms in symmetric closed monoidal categories with additional structure. This is made explicit by showing how to represent proofs in linear logic as linear maps between vector spaces. The interesting part of this vector space semantics is based on the cofree cocommutative coalgebra of Sweedler.
This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $\mathbf{E}_{\infty}$-ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms $TMF$ and the non-periodic and non-connective $Tmf$. We find that $\mathrm{Pic} (TMF)$ is cyclic of order 576, generated by the suspension $ΣTMF $ (a result originally due to Hopkins), while $\mathrm{Pic}(Tmf) = \mathbb{Z}\oplus \mathbb{Z}/24$. In particular, we show that there exists an invertible $Tmf$-module which is not equivalent to a suspension of $Tmf$.
In this paper we propose and study few applications of the base structured categories $\mathcal{X} \rtimes_{\mathbf{F}} \mathbf{C}$, $\int_{\mathbf{C}} \bar{\mathbf{F}}$, $\mathcal{X} \rtimes_{\mathbb{F}} \mathbf{C}$ and ${\int_{\mathbf{C}} \bar{\mathbb{F}}}$. First we show classic transformation groupoid $X /\!\!/ G$ simply being a base-structured category ${\int_{\mathbf{G}} \bar{F}}$. Then using permutation action on a finite set, we introduce the notion of a hierarchy of base structured categories $[(\mathcal{X}_{2a} \rtimes_{\mathbf{F_{2a}}} \mathbf{B}_{2a}) \amalg (\mathcal{X}_{2b} \rtimes_{\mathbf{F_{2b}}} \mathbf{B}_{2b}) \amalg ...] \rtimes_{\mathbf{F_{1}}} \mathbf{B}_1$ that models local and global structures as a special case of composite Grothendieck fibration. Further utilizing the existing notion of transformation double category $(\mathcal{X}_{1} \rtimes_{\mathbf{F_{1}}} \mathbf{B}_{1}) /\!\!/ \mathbf{2G}$, we demonstrate that a hierarchy of bases naturally leads one from 2-groups to n-category theory. Finally we prove that every classic Klein geometry is the Grothendieck completion ($\mathbf{G} = \mathcal{X} \rtimes_{\mathbb{F}} \mathbf{H}$) of ${\mathbb{F}}: \mathb
Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that $φ: X \rightarrow X$ is a morphism of $\mathscr{C}$ with core inverse $φ^{\co} : X \rightarrow X$ and $η: X \rightarrow X$ is a morphism of $\mathscr{C}$ such that $1_X+φ^{\co}η$ is invertible. Let $α=(1_X+φ^{\co}η)^{-1},$ $β=(1_X+ηφ^{\co})^{-1},$ $\varepsilon=(1_X-φφ^{\co})ηα(1_X-φ^{\co}φ),$ $γ=α(1_X-φ^{\co}φ)β^{-1}φφ^{\co}β,$ $σ=αφ^{\co}φα^{-1}(1_X-φφ^{\co})β,$ $δ=β^{\ast}(φ^{\co})^{\ast}η^{\ast}(1_X-φφ^{\co})β.$ Then $f=φ+η-\varepsilon$ has a core inverse if and only if $1_X-γ$, $1_X-σ$ and $1_X-δ$ are invertible. Moreover, the expression of the core inverse of $f$ is presented. Let $R$ be a unital $\ast$-ring and $J(R)$ its Jacobson radical, if $a\in R^{\co}$ with core inverse $a^{\co}$ and $j\in J(R)$, then $a+j\in R^{\co}$ if and only if $(1-aa^{\co})j(1+a^{\co}j)^{-1}(1-a^{\co}a)=0$. We also give the similar results for the dual core inverse.
We use non-standard analysis to define a category $^\star\!\operatorname{Hilb}$ suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and we show that standard bounded operators can be suitably embedded in it. We show the existence of unital special commutative $\dagger$-Frobenius algebras, and we conclude $^\star\!\operatorname{Hilb}$ to be compact closed, with partial traces and a Hilbert-Schmidt inner product on morphisms. We exemplify our techniques on the textbook case of 1-dimensional wavefunctions with periodic boundary conditions: we show the momentum and position observables to be well defined, and to give rise to a strongly complementary pair of unital commutative $\dagger$-Frobenius algebras.
Furber and Jacobs have shown in their study of quantum computation that the category of commutative C*-algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C*-algebras with MIU-maps (linear maps which preserve multiplication, involution and unit). [Furber and Jacobs, 2013] In this paper, we prove a non-commutative variant of this result: the category of C*-algebras and PU-maps is isomorphic to the Kleisli category of a comonad on the subcategory of MIU-maps. A variation on this result has been used to construct a model of Selinger and Valiron's quantum lambda calculus using von Neumann algebras. [Cho and Westerbaan, 2016]
In this paper, we deal with quantum theories on presheaves and sheaves on context categories consisting of commutative von Neumann algebras of bounded operators on a Hilbert space, from two viewpoints. One is to reduce presheaf-based topos quantum theory via sheafification, and the other is to import quantum probabilities to the reduced sheaf quantum theory. The first is done by means of a functor that selects some expedient contexts. It defines a Grothendieck topology on the category consisting of all contexts, hence, induces a sheaf topos on which we construct a downsized quantum theory. Also, we show that the sheaf quantum theory can be replaced by an equivalent, more manageable presheaf quantum theory. Quantum probabilities are imported by means of a Grothendieck topology that is defined on a category consisting of probabilities and enables to regard them as intuitionistic truth-values. From these topologies, we construct another Grothendieck topology that is defined on the product of the context category and the probability category and reflects the selection of contexts and the identification of probabilities with truth-values. We construct a quantum theory equipped with quan
We introduce a class of toposes called "absolutely locally compact" toposes and of "admissible" sheaf of rings over such toposes. To any such ringed topos $(\mathcal{T},A)$ we attach an involutive convolution algebra $\mathcal{C}_c(\mathcal{T},A)$ which is well defined up to Morita equivalence and characterized by the fact that the category of non-degenerate modules over $\mathcal{C}_c(\mathcal{T},A)$ is equivalent to the category of sheaf of $A$-module over $\mathcal{T}$. In the case where $A$ is the sheaf of real or complex Dedekind numbers, we construct several norms on this involutive algebra that allows to complete it in various Banach and $C^*$-algebras: $L^1(\mathcal{T},A)$, $C^*_{red}(\mathcal{T},A)$ and $C^*_{max}(\mathcal{T},A)$. We also give some examples where this construction corresponds to well known constructions of involutive algebras, like groupoids convolution algebra and Leavitt path algebras.
We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal V$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category $\mathcal T$. Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather satisfy a braided interchange relation. Strikingly, our classification is slightly more general than what one might have anticipated in terms of strong monoidal functors $\mathcal V \to Z(\mathcal T)$. We would like to understand this further; in a future paper we show that the functor is strong if and only if the enriched category is `complete' in a certain sense. Nevertheless it remains to understand what non-complete enriched categories may look like. One should think of our construction as a generalization of de-equivariantization, which takes a strong monoidal functor $\mathsf{Rep}(G) \t
We investigate notions of ambiguity and partial information in categorical distributional models of natural language. Probabilistic ambiguity has previously been studied using Selinger's CPM construction. This construction works well for models built upon vector spaces, as has been shown in quantum computational applications. Unfortunately, it doesn't seem to provide a satisfactory method for introducing mixing in other compact closed categories such as the category of sets and binary relations. We therefore lack a uniform strategy for extending a category to model imprecise linguistic information. In this work we adopt a different approach. We analyze different forms of ambiguous and incomplete information, both with and without quantitative probabilistic data. Each scheme then corresponds to a suitable enrichment of the category in which we model language. We view different monads as encapsulating the informational behaviour of interest, by analogy with their use in modelling side effects in computation. Previous results of Jacobs then allow us to systematically construct suitable bases for enrichment. We show that we can freely enrich arbitrary dagger compact closed cate
Effect algebras are one of the generalizations of Boolean algebras proposed in the quest for a quantum logic. Frobenius algebras are a tool of categorical quantum mechanics, used to present various families of observables in abstract, often nonstandard frameworks. Both effect algebras and Frobenius algebras capture their respective fragments of quantum mechanics by elegant and succinct axioms; and both come with their conceptual mysteries. A particularly elegant and mysterious constraint, imposed on Frobenius algebras to characterize a class of tripartite entangled states, is the antispecial law. A particularly contentious issue on the quantum logic side is the modularity law, proposed by von Neumann to mitigate the failure of distributivity of quantum logical connectives. We show that, if quantum logic and categorical quantum mechanics are formalized in the same framework, then the antispecial law of categorical quantum mechanics corresponds to the natural requirement of effect algebras that the units are each other's unique complements; and that the modularity law corresponds to the Frobenius condition. These correspondences lead to the equivalence announced in the title. Ali
Techniques from higher categories and higher-dimensional rewriting are becoming increasingly important for understanding the finer, computational properties of higher algebraic theories that arise, among other fields, in quantum computation. These theories have often the property of containing simpler sub-theories, whose interaction is regulated in a limited number of ways, which reveals a topological substrate when pictured by string diagrams. By exploring the double nature of computads as presentations of higher algebraic theories, and combinatorial descriptions of "directed spaces", we develop a basic language of directed topology for the compositional study of algebraic theories. We present constructions of computads, all with clear analogues in standard topology, that capture in great generality such notions as homomorphisms and actions, and the interactions of monoids and comonoids that lead to the theory of Frobenius algebras and of bialgebras. After a number of examples, we describe how a fragment of the ZX calculus can be reconstructed in this framework.
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. The titular category has nice formal properties: it is bicomplete and is a symmetric monoidal category, with monoidal product closely related to the Boardman-Vogt tensor product of operads. Tools developed in this article, which is the first part of a larger work, include a generalized version of multilinearity of functors, a free prop construction defined on certain "generalized graphs", and the relationship between the category of props and the categories of permutative categories and of operads.
We study nilpotency in the context of exact Mal'tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie's functor calculus. We show in particular that the reflection into the subcategory of $n$-nilpotent objects is the universal endofunctor of degree $n$ if and only if every $n$-nilpotent object is $n$-folded. In the special context of a semi-abelian category, an object is $n$-folded precisely when its Higgins commutator of length $n+1$ vanishes.
It has been proved by Bergh and Thaule that the higher mapping cone axiom is equivalent to the higher octahedral axiom for n-angulated categories. In this note, we use homotopy cartesian diagrams to give several new equivalent statements of the higher mapping cone axiom, which are applied to explain the higher octahedral axiom.
We introduce the class of isotypic $2$-representations for finitary $2$-categories and the notion of inflation of $2$-representations. Under some natural assumptions we show that isotypic $2$-representations are equivalent to inflations of cell $2$-representations.
We characterise regular Goursat categories through a specific stability property of regular epimorphisms with respect to pullbacks. Under the assumption of the existence of some pushouts this property can be also expressed as a restricted Beck-Chevalley condition, with respect to the fibration of points, for a special class of commutative squares. In the case of varieties of universal algebras these results give, in particular, a structural explanation of the existence of the ternary operations characterising $3$-permutable varieties of universal algebras.
In this article, we define and study the class of simple transitive $2$-representations of finitary $2$-categories. We prove a weak version of the classical Jordan-Hölder Theorem where the weak composition subquotients are given by simple transitive $2$-representations. For a large class of finitary $2$-categories we prove that simple transitive $2$-representations are exhausted by cell $2$-representations. Finally, we show that this large class contains finitary quotients of $2$-Kac-Moody algebras.
We classify simple transitive $2$-representations of certain $2$-sub\-ca\-te\-go\-ri\-es of the $2$-category of Soergel bimodules over the coinvariant algebra in Coxeter types $B_2$ and $I_2(5)$. In the $I_2(5)$ case it turns out that simple transitive $2$-representations are exhausted by cell $2$-representations. In the $B_2$ case we show that, apart from cell $2$-representations, there is a unique, up to equivalence, additional simple transitive $2$-representation and we give an explicit construction of this $2$-representation.
For a pivotal finite tensor category $\mathcal{C}$ over an algebraically closed field $k$, we define the algebra $\mathsf{CF}(\mathcal{C})$ of class functions and the internal character $\mathsf{ch}(X) \in \mathsf{CF}(\mathcal{C})$ for an object $X \in \mathcal{C}$ by using an adjunction between $\mathcal{C}$ and its monoidal center $\mathcal{Z}(\mathcal{C})$. We also develop the integral theory in a unimodular finite tensor category by using the same adjunction. By utilizing these tools, we extend some results in the character theory of finite-dimensional Hopf algebras to this category-theoretical setting. Our main result is that the map $\mathsf{ch}: \mathsf{Gr}_k(\mathcal{C}) \to \mathsf{CF}(\mathcal{C})$ given by taking the internal character is a well-defined injective algebra map, where $\mathsf{Gr}_k(\mathcal{C})$ is the scalar extension of the Grothendieck ring of $\mathcal{C}$ to $k$. Moreover, under the assumption that $\mathcal{C}$ is unimodular, the map $\mathsf{ch}$ is an isomorphism if and only if $\mathcal{C}$ is semisimple. As an application, we show that the algebra $\mathsf{Gr}_{k}(\mathcal{C})$ is semisimple if $\mathcal{C}$ is a non-degenerate pivotal fusion cat
In this short note, a topos - called the topos of the connectivity space - is associated with every such space.
Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences. Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories. The definition has an intrinsic non-symmetry. Suppose that we have a triangulated category in which idempotents split and either for which the endomorphism rings of all objects are artinian, or which is the category of compact objects in an algebraic compactly generated triangulated K-category. Then we show that the non-symmetry in the algebraic definition of the degeneration is inessential in the sense that the two possible choices which can be made in the definition lead to the same concept.
Separable Frobenius monoidal functors were defined and studied under that name by Szlachanyi and by Day and Pastro, and in a more general context by Cockett and Seely. Our purpose here is to develop their theory in a very precise sense. We determine what kinds of equations in monoidal categories they preserve. For example we show they preserve lax (meaning not necessarily invertible) Yang-Baxter operators, weak Yang-Baxter operators in the sense of Alonso Alvarez et al., and (in the braided case) weak bimonoids in the sense of Pastro and Street. In fact, we characterize which monoidal expressions are preserved (or rather, are stable under conjugation in a well-defined sense). We show that every weak Yang-Baxter operator is the image of a genuine Yang-Baxter operator under a separable Frobenius monoidal functor. Prebimonoidal functors are also defined and discussed.
We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for infinity-operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. In an appendix, we explain how our arguments can be used to extend the results of Cisinski, giving the existence of minimal fibrations in model categories of presheaves over generalised Reedy categories of a rather common type. Besides some applications to the theory of algebras over infinity-operads, we also prove a gluing result for parametrized connective spectra (or Gamma-spaces).
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Wei Wang contributes to research discovery and scholarly infrastructure.
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