In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete “operational” approach, (...) in which the states and measurement outcomes associated with a physical system are represented in terms of what we here call a convex operational model: a certain dual pair of ordered linear spaces–generally, not isomorphic to one another. On the other hand, state spaces for which there is such an isomorphism, which we term weakly self-dual, play an important role in reconstructions of various quantum-information theoretic protocols, including teleportation and ensemble steering. In this paper, we characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol—hence, as a form of classical probabilistic conditioning. In a large class of cases, which includes both the classical and quantum cases, the relevant compact closed categories are degenerate, in the weak sense that every object is its own dual. We characterize the dagger-compactness of such a category (with respect to the natural adjoint) in terms of the existence, for each system, of a symmetric bipartite state, the associated conditioning map of which is an isomorphism. (shrink)

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate (...) class='Hi'>compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropie quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of 'conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category. (shrink)

The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic _in re_ interpretation of mathematical structuralism. In each context, what we (...) aim to show is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning. (shrink)

Within the context of an involutive monoidal category the notion of a comparison relation ${\mathsf{cp} : \overline{X} \otimes X \rightarrow \Omega}$ is identified. Instances are equality = on sets, inequality ${\leq}$ on posets, orthogonality ${\perp}$ on orthomodular lattices, non-empty intersection on powersets, and inner product ${\langle {-}|{-} \rangle}$ on vector or Hilbert spaces. Associated with a collection of such (symmetric) comparison relations a dagger category is defined with “tame” relations as morphisms. Examples include familiar categories in the (...) foundations of quantum mechanics, such as sets with partial injections, or with locally bifinite relations, or with formal distributions between them, or Hilbert spaces with bounded (continuous) linear maps. Of one particular example of such a dagger category of tame relations, involving sets and bifinite multirelations between them, the categorical structure is investigated in some detail. It turns out to involve symmetric monoidal dagger structure, with biproducts, and dagger kernels. This category may form an appropriate universe for discrete quantum computations, just like Hilbert spaces form a universe for continuous computation. (shrink)

We prove a structure theorem for compact inverse categories. The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for inverse monoids: they are inductive groupoids. A particularly nice case due to Clifford is that commutative inverse monoids become semilattices of abelian groups. It has also been categorified by Hoehnke and DeWolf-Pronk to a structure theorem for inverse categories as locally complete inductive groupoids. We show that in the case of compact inverse categories, this takes the particularly nice form of a (...) semilattice of compact groupoids. Moreover, one-object compact inverse categories are exactly commutative inverse monoids. Compact groupoids, in turn, are determined in particularly simple terms of 3-cocycles by Baez-Lauda. (shrink)

We discuss issues related to constructing an orthomodular structure from an object in a category. In particular, we consider axiomatics related to Baer *-semigroups, partial semigroups, and various constructions involving dagger categories, kernels, and biproducts.

In this paper, we discuss some questions about compactness in MV-topological spaces. More precisely, we first present a Tychonoff theorem for such a class of fuzzy topological spaces and some consequence of this result, among which, for example, the existence of products in the category of Stone MV-spaces and, consequently, of coproducts in the one of limit cut complete MV-algebras. Then we show that our Tychonoff theorem is equivalent, in ZF, to the Axiom of Choice, classical Tychonoff theorem, and (...) Lowen’s analogous result for lattice-valued fuzzy topology. Last, we show an extension of the Stone–Čech compactification functor to the category of MV-topological spaces, and we discuss its relationship with previous works on compactification for fuzzy topological spaces. (shrink)

In this paper we define sheaf spaces of BL-algebras (or BL-sheaf spaces), we study completely regular and compact BL-sheaf spaces and compact representations of BL-algebras and, finally, we prove that the category of non-trivial BL-algebras is equivalent with the category of compact local BL-sheaf spaces.

Let A be the category of all reduced compact complex spaces, viewed as a multi-sorted first order structure, in the standard way. Let U be a sub-category of A, which is closed under the taking of products and analytic subsets, and whose morphisms include the projections. Under the assumption that Th(U) is unidimensional, we show that Morley rank is equal to Noetherian dimension, in any elementary extension of U. As a result, we are able to show that (...) Morley degree is definable in Th(U), when Th(U) is unidimensional. (shrink)

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with (...) the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces. (shrink)

In this paper I present an empirical solution to the puzzle of Macbeth's dagger. The puzzle of Macbeth's dagger is the question of whether, in having his fatal vision of a dagger, Macbeth sees a dagger. I answer this question by addressing a more general one: the question of whether perceptual verbs are intensional transitive verbs (ITVs). I present seven experiments, each of which tests a collection of perceptual verbs for one of the three features characteristic (...) of ITVs. One of these features is Nonexistence: the failure of sentences involving transitive verbs to entail the existence of their direct objects. The experiments reveal that with respect to all three of these features, "see" behaves much more like a paradigmatically extensional verb than an intensional one. But surprisingly, unlike "see", "perceive" behaves much more like a paradigmatically intensional verb. This shows that the category of perceptual verbs is not uniform with respect to the features of intensionality; while Macbeth does not see a dagger, he may still perceive one. (shrink)

We prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not (...) dually equivalent to any finitely accessible category, any first-order definable class of structures, and any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.Abstract taken directly from the thesis.E-mail: [email protected]: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf. (shrink)

CSR has become an important element in the business strategy of a growing number of companies worldwide. A large number of initiatives have been developed that aim to support companies in developing, implementing, and communicating about CSR. The Global Compact (GC), initiated by the United Nations, stands out. Since its launch in 2000, it has grown to about 2900 companies and 3800 members in total. The GC combines several mechanisms to support CSR strategies: normative principles, networks for learning and (...) co-operation, and communication and transparency about CSR activities. However, up to now only a few empirical evaluations of the contribution of the GC to CSR strategies have been conducted that however have not differentiated between different types of companies (regarding type of industry or regarding the maturity of CSR). This paper aims to partly fill this knowledge gap by a case study examination of three frontrunner companies in the telecommunications industry. The results show that the GC is only one of the many initiatives that these companies employ in shaping, implementing, and reporting about their CSR strategies, and that its role is at most modest. There are two important reasons. One is that many of the CSR issues that these companies deal with are industry specific and are hence addressed in specific networks. The second reason is that the GC principles are perceived as minimum requirements that do not provide many incentives to the three case study companies to perform better. A differentiation of norms for GC members is expected to enhance the contribution of the GC to CSR strategy employment, not only for frontrunner companies but as well for other categories of companies. (shrink)

The locale corresponding to the real interval [ − 1, 1] is an interval object, in the sense of Escardó and Simpson, in the category of locales. The map, mapping a stream s of signs ±1 to, is a proper localic surjection; it is also expressed as a coequalizer. The proofs are valid in any elementary topos with natural numbers object.

Working in a fixed Grothendieck topos Sh(C, J C ) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, J C ). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\) , such as the downward Löwenheim–Skolem theorem, the completeness theorem (...) and Barwise compactness. (shrink)

Dagger argues for a republican liberalism that, while celebrating the liberal heritage of autonomy and rights, solidly places these within social relations and obligations, which while ubiquitous, are often obscured and forgotten.

At the UN, important projects laying down transnational corporations' (TNCs) human rights responsibilities have been launched without ever clarifying the relevant theoretical foundations. One of the consequences is that the human rights principles in projects like the 2000 UN Global Compact and the 2003 Norms on the Responsibilities of Transnational Corporations and Other Business Enterprises with Regard to Human Rights can be understood in different ways, which should not cause surprise given that their authors come from diverse backgrounds, including (...) economics and public policy. An examination of these projects and the views of their authors reveals that, although they are superficially linked to international human rights law, they go well beyond it and attempt to deal with corporate social responsibility issues in ways that elude neat classification as fitting neatly in either legal or non-legal categories. Too little attention has been paid to how in the course of developing these projects the legal and ethical dimensions have become entwined and how lines have gotten blurred. Meanwhile, there has been recognition that these UN projects have emerged simply as ad hoc responses to practical concerns about the sustainability of globalization. The lack of any foundational theory or normative framework should be addressed; it is time to bring together specialists from different fields concerned with the human rights responsibilities of corporations to see if it is possible to define a coherent overarching theory for these UN projects. (shrink)

It is clear that a revival of republicanism is under way, but it is not clear that the republican tradition truly speaks to contemporary concerns. In particular, it is not clear that republicanism has anything of value to say about economic matters in the early 21st century. I respond to this worry by delineating the main features of a neo-republican civic economy that is, I argue, reasonably coherent and attractive. Such an economy will preserve the market, while constraining it to (...) serve public purposes, and promote what John Rawls calls a ‘property-owning democracy’. To accomplish these ends, a civic economy is likely to concern itself with the character of work and the workplace, to take steps to preserve and protect the sense of community or publicity, to levy an inheritance tax and a progressive consumption tax, and to provide some kind of ‘social’ or ‘civic’ minimum of support to all citizens. Key Words: republicanism • equality • self-government • deliberative politics • civic virtue • publicity • markets • property-owning democracy • basic income • inheritance tax • consumption tax. (shrink)

We show that the statement “separable, countably compact, regular spaces are Baire” is deducible from a strictly weaker form than AC, namely, CAC . We also find some characterizations of the axiom of dependent choices.

In Playing Fair, Richard Dagger provides a unified theory of political obligation and the justification of punishment that takes its bearings from the principle of fair play. Dagger argues that members of a just polity have an obligation to obey its laws because they have an obligation of reciprocity or fair play to one another.

This article defends the fair-play theory of legal punishment against three objections. The first, the irrelevance objection, is the long-standing complaint that fair play fails to capture what it is about crimes that makes criminals deserving of punishment ; the others are the recently raised false-equivalence and lacks-integration objections. In response, I sketch an account of fair-play theory that is grounded in a conception of the political order as a meta- cooperative practice—a conception that falls somewhere between contractual and communitarian (...) conceptions—and draw on this account to show how the theory can overcome the objections. (shrink)

The Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Čech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the "Henkin method" of proving deductive completeness of logical systems. The Rasiowa-Sikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces.

We introduce the framework of AECats, generalizing both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of models of a positive or continuous theory is an AECat. The Kim–Pillay theorem for first-order logic characterizes simple theories by the properties dividing independence has. We prove a version of (...) the Kim–Pillay theorem for AECats with the amalgamation property, generalizing the first-order version and existing versions for positive logic. (shrink)

ABSTRACTAccording to the standard or traditional account, those who hold political authority legitimately have a right to rule that entails an obligation of obedience on the part of those who are subject to their authority. In recent decades, however, and in part in response to philosophical anarchism, a number of philosophers have challenged the standard account by reconceiving authority in ways that break or weaken the connection between political authority and obligation. This paper argues against these revisionist accounts in two (...) ways: first, by pointing to defects in their conceptions of authority; and second, by sketching a fair-play approach to authority and political obligation that vindicates the standard account. (shrink)

The identification and positioning of sacred space within contemporary contexts has, to date, received scant attention. In reflecting upon a broad spectrum of conceptions of what constitutes sacred space, this collection of interdisciplinary essays presents a new perspective on an area that is developing into an important theological and philosophical concept.

In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example, linear independence in vector spaces yields such a relation.Some important classes in the stability hierarchy are stable, simple, and NSOP $_1$, each being contained in the next. For each of (...) these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example, simplicity is equivalent to admitting a certain independence relation, which must then be unique.All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic, and even a very general category-theoretic framework. In this thesis we continue this work.We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple, or NSOP $_1$ -like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing, and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.Switching frameworks, we generalise Kim-dividing in NSOP $_1$ theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP $_1$ if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.Abstract prepared by Mark Kamsma.E-mail:[email protected]:https://markkamsma.nl/phd-thesis. (shrink)

Republicanism may seem to be a nostalgic politics of place that is incapable of responding to the challenges of globalization.The burden of this essay is to demonstrate that this view is both right and wrong - right in regarding republicanism as a politics of place, butwrong in thinking that such a form of politics is irrelevant to an increasingly interconnected world. On the contrary, the republican concern for place provides the basis for the responsible, public-spirited action that cosmopolitan theorists need (...) to sustain their visions of global politics. (shrink)

One need not look far beyond the titles and distinctive phrases to find a deep and abiding concern for civic virtue in John Rawls’ writings. This chapter provides the necessary account of civic virtue and Rawls's conception of it. For this, it relies most heavily on Rawls's last book, Justice as Fairness: A Restatement. The importance of concepts of citizenship and civility is more evident in Justice as Fairness, than in Rawls's other books. According to Rawls, the most‐advantaged members of (...) a just society would demonstrate their commitment to public culture by inhibiting the wastes of endless self‐ and group‐interested bargaining and offering some hope of realizing social concord and civic friendship. Beyond encouraging civic engagement, Rawls's “political liberalism” has many points of contact with the republican tradition. One such point of contact arises when he indicates the importance to a well‐ordered society of civic education. (shrink)

This chapter contains sections titled: Liberals versus Communitarians? Individualism Community From Community to Republic Acknowledgments Notes References.