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1.
This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely
the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest,
such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have
interesting categorical/logical/order-theoretic properties, in terms of kernel fibrations, such as existence of pullbacks,
factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and and-then connectives are obtained,
as adjoints, via the existential-pullback adjunction between fibres. 相似文献
2.
M.A. Batanin 《Advances in Mathematics》2008,217(1):334-385
The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A. Joyal and R. Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category.In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckmann-Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of n-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Symn from a certain subcategory of n-operads to the category of symmetric operads such that the category of one object, one arrow, … , one (n−1)-arrow algebras of A is isomorphic to the category of algebras of Symn(A). Under some mild conditions, we present an explicit formula for Symn(A) which involves taking the colimit over a remarkable categorical symmetric operad.We will consider some applications of the methods developed to the theory of n-fold loop spaces in the second paper of this series. 相似文献
3.
Crossed Modules and Quantum Groups in Braided Categories 总被引:2,自引:0,他引:2
Yu. N. Bespalov 《Applied Categorical Structures》1997,5(2):155-204
Let A be a Hopf algebra in a braided category
. Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category
of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group
the corresponding braided category of modules
is identified with a full subcategory in
. The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid–Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized. 相似文献
4.
This paper investigates dagger kernel categories which are considered first by Crown (J Nat Sci Math 15:11?C25, 1975) and used by Heunen and Jacobs (Order 27:177?C212, 2010) in their study of quantum logic from the perspective of categorical logic. The inverse Baer*-categories with splitting projections as special dagger kernel categories have a central place in our investigations. The inverse Baer*-categories with splitting and closed projections are Boolean and therefore the subobject lattices of such categories are representing classical logics. Examples are presented at every stage of our investigations. 相似文献
5.
We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka–Krein reconstruction problem. We show that every concrete semisimple tensor *-category with conjugates is equivalent to the category of finite-dimensional nondegenerate *-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka–Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well-known general result concerning discrete multiplier Hopf *-algebras. 相似文献
6.
A. Van Daele 《代数通讯》2013,41(6):2341-2386
A simple and nice structure theorem for orthogroups was given by Petrich in 1987. In this paper, we consider a generalized orthogroup, that is, a quasi-completely regular semigroup with a band of idempotents in which its set of regular elements, namely, RegS, forms an ideal of S. A method of construction of such semigroups is provided and as a result, the Petrich structure theorem of orthogroups becomes an immediate corollary of our theorem on generalized orthogroups. An example of such generalized orthogroup is also constructed. This example provides some useful information for the construction of various kinds of quasi-completely regular semigroups. 相似文献
7.
Ukrainian Mathematical Journal - We prove a Hinich-type theorem on the existence of a model structure on a category related by adjunction to the category of differential graded modules over a... 相似文献
8.
We generalize results on existence of recollement situations of singularity categories of lower triangular Gorenstein algebras and stable monomorphism categories of Cohen–Macaulay modules. 相似文献
9.
Marco Grandis 《Applied Categorical Structures》2007,15(4):341-353
Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space’ has directed homotopies
(generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories.
Applications have been mostly developed in the theory of concurrency. Unexpected links with noncommutative geometry and the
modelling of biological systems have emerged.
Work partially supported by MIUR Research Projects. 相似文献
10.
Let 𝒳 ? 𝒜 be subcategories of a triangulated category 𝒯, and 𝒳 a functorially finite subcategory of 𝒜. If 𝒜 has the properties that any 𝒳-monomorphism of 𝒜 has a cone and any 𝒳-epimorphism has a cocone, then the subfactor category 𝒜/[𝒳] forms a pretriangulated category in the sense of [4]. Moreover, the above pretriangulated category 𝒜/[𝒳] with 𝒯(𝒳, 𝒳[1]) = 0 becomes a triangulated category if and only if (𝒜, 𝒜) forms an 𝒳-mutation pair and 𝒜 is closed under extensions. 相似文献
11.
We present a new characterisation of Goursat categories in terms of special kind of pushouts, that we call Goursat pushouts. This allows one to prove that, for a regular category, the Goursat property is actually equivalent to the validity of the
denormalised 3-by-3 Lemma. Goursat pushouts are also useful to clarify, from a categorical perspective, the existence of the
quaternary operations characterising 3-permutable varieties. 相似文献
12.
Clemens Berger 《Advances in Mathematics》2002,169(1):118-175
13.
14.
Amihud Gilead 《Metaphysica》2008,9(1):1-16
Considering various arguments in Hume’s Treatise, I reconstruct a Humean argument against personal identity or unity. According to this argument, each distinct perception is separable from the bundle of perceptions to which it belongs and is thus transferable either to the external, material reality or to another psychical reality, another bundle of perceptions. Nevertheless, such transference (Hume’s word!) is entirely illegitimate, otherwise Hume’s argument against causal inference would have failed; furthermore, it violates private, psychical accessibility. I suggest a Humean thought experiment clearly demonstrating that, to the extent that anything within a psychical reality is concerned, no distinction leads to separation or transference and that private, psychical accessibility has to be allowed in the Humean argument for personal identity or unity. Private accessibility and psychical untransferability secure personal identity and unity. Referring to the phenomenon of multiple personality along the lines of the Humean argument for personal identity or unity, I illustrate both private accessibility and a possible notion of one and the same person distinct from his/her alters or psychical parts. Finally, I show why Parfit’s Humean argument against personal identity must fail. 相似文献
15.
We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories
possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property
F if the associated braid group representations factor over a finite group, and suggest that categories of integral Frobenius-Perron
dimension are precisely those with property F. 相似文献
16.
Peter Arndt Rodrigo de Alvarenga Freire Odilon Otavio Luciano Hugo Luiz Mariano 《Logica Universalis》2007,1(1):3-39
We explore the possibility and some potential payoffs of using the theory of accessible categories in the study of categories
of logics. We illustrate this by two case studies focusing on the category of finitary structural logics and its subcategory
of algebraizable logics.
Mathematics Subject Classification (2000): Primary 03B22; Secondary 18C35. 相似文献
17.
18.
We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories. 相似文献
19.
Zhang Shunhua 《东北数学》1994,(3)
V-modulesandV-additiveCategoriesZhangShunhua(张顺华)(DepartmentofMathematics,BeijingNormalUniversity,Beijing,100875)Abstract:The... 相似文献
20.
Let G{{\mathcal G}} be a group, Λ a G{{\mathcal G}}-graded Artin algebra and gr(Λ) denote the category of finitely generated G{{\mathcal G}}-graded Λ-modules. This paper provides a framework that allows an extension of tilting theory to Db(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and to study connections between the tilting theories of Db(L){{\mathcal D}}^b(\Lambda) and Db(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)). In particular, using that if T is a gradable Λ-module, then a grading of T induces a G{{\mathcal G}}-grading on EndΛ(T), we obtain conditions under which a derived equivalence induced by a gradable Λ-tilting module T can be lifted to a derived equivalence between the derived categories Db(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and Db(gr(EndL(T))){{\mathcal D}}^b(\rm gr(\rm End_{\Lambda}(\textit T))). 相似文献