Generalized Enrichments of Categories for Operads

2007 ◽  
Vol 14 (01) ◽  
pp. 61-78
Author(s):  
Zizhu Ma
Keyword(s):  
Monoidal Category ◽  
Special Kind ◽  
Category Structure ◽  
Commutative Monoids ◽  
Enriched Categories

Most enriched categories also have an ordinary category structure which is compatible with the enrichment on them. In this paper, enrichments in a monoidal category are generalized to arbitrary categories. These specialize to the classical enrichments when sets are regraded as discrete categories. We also generalize the definitions of PROs and PROPs as some generalized enrichments of categories. Then an operad in some monoidal category corresponds to a generalized PROP. Algebras of operads induce some special kind of monoidal functors. In the category of small categories, we construct several operads to define lax monoids and lax commutative monoids which are formal descriptions of natural associativity and commutativity. Using this identification, operads and their algebras can be studied by lax commutative monoids and morphisms between them.

scholarly journals Monoidal Categories Enriched in Braided Monoidal Categories

2017 ◽  
Vol 2019 (11) ◽  
pp. 3527-3579 ◽  
Author(s):  
Scott Morrison ◽  
David Penneys
Keyword(s):  
Monoidal Category ◽  
Cartesian Product ◽  
Basic Theory ◽  
Monoidal Categories ◽  
Enriched Category ◽  
Enriched Categories ◽  
Monoidal Functor ◽  
Natural Transformations ◽  
Braided Monoidal Categories ◽  
Set Up

Abstract 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 centre 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 article, 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 ${\mathsf {Rep}}(G) \to Z(\mathcal{T})$ for some finite group $G$ and a monoidal category $\mathcal{T}$, and produces a new monoidal category $\mathcal{T} _{{/\hspace{-2px}/}G}$. In our setting, given any braided oplax monoidal functor $\mathcal{V} \to Z(\mathcal{T})$, for any braided $\mathcal{V}$, we produce $\mathcal{T} _{{/\hspace{-2px}/}\mathcal{V}}$: this is not usually an ‘honest’ monoidal category, but is instead $\mathcal{V}$-enriched. If $\mathcal{V}$ has a braided lax monoidal functor to ${\mathsf {Vec}}$, we can use this to reduce the enrichment to ${\mathsf {Vec}}$, and this recovers de-equivariantization as a special case. This is the published version of arXiv:1701.00567.

Research on Starvation in the Warsaw Ghetto – Ethical Problems

2010 ◽  
pp. 439-450
Author(s):  
Marta Janczewska
Keyword(s):  
Research Team ◽  
Ethical Dilemmas ◽  
Medical Profession ◽  
Research Work ◽  
Special Kind ◽  
Scientific Discipline ◽  
Medical Treatments ◽  
Warsaw Ghetto ◽  
Dying Patients ◽  
To Receive

Research team of physicians and lab technicians under Izrael Milejkowski’s direction undertook the effort to carry out a series of clinical and biochemical experiments on patients dying of starvation in the Warsaw ghetto so as to receive the fullest possible picture of hunger disease. The research was carried out according to all the rigors of strict scientific discipline, and the authors during their work on academic articles, published it after the war entitled: „Starvation disease: hunger research carried out in the Warsaw ghetto in 1942,” according to their own words, they “supplemented the gap in accordance with the progress of knowledge.” The article is devoted to the reflections over ethical dilemmas of the research team, who were forced in their work to perform numerous medical treatments of experimental nature on extremely exhausted patients. The ill, according to Dr Fajgenblat’s words,“demonstrated negativism toward the research and treatment, which extremely hindered the work, and sometimes even frustrated it.” The article attempts to look at the monumental research work of the Warsaw ghetto doctors as a special kind of response of the medical profession to the feeling of helplessness to the dying patients. The article analyzes the situation of Warsaw ghetto doctors, who undertook the research without support of any outer authority, which could settle their possible ethical dilemmas (Polish deontological codes, European discussions on the conditions of the admissibility of medical research on patients, etc.).

The Evolution of Psychoanalytic Practice with Psychotic Patients

2002 ◽  
Vol 4 (1) ◽  
pp. 31-43
Author(s):  
Sophie de Mijolla-Mellor
Keyword(s):  
Special Kind ◽  
Main Hypothesis ◽  
Actual Case ◽  
Definition Of ◽  
Psychoanalytic Method

This paper deals with the evolution of the psychoanalytic practice with psychotic patients beginning with Freud's scepticism about the transference capacities of those patients to a new definition of a special kind of psychotic transference. The main hypothesis is that the actual case of psychotics within a psychoanalytic cure has modified the psychoanalytic method itself, even in the field of neuroses. Within the framework and, more specifically, in the case of schizophrenics, this paper develops some reflections on the evolution of the three following concepts: transference/countertransference, communication and interpretation, and reality.

Metaphorik im Kommunikationsbereich der Sportfachsprachlichkeit

Fachsprache ◽  
2019 ◽  
Vol 41 (S1) ◽  
pp. 87-100
Author(s):  
Wenke Mückel
Keyword(s):  
Olympic Games ◽  
Special Kind ◽  
European Football ◽  
Sport Events ◽  
The Olympic Games

Metaphorical elements are a highly productive language means in live reports about sport events on TV. They occur in different relations to what is simultaneously seen on screen and depend on the reporter as well as on the special kind of sport. But nevertheless, general structures and functions of metaphors in those medium-bound oral texts can be indicated; as one of the markers they contribute to what is often called language of sport or maybe rather communicative template of sport. Examples taken from TV reports of the European Football Championship and the Olympic Games (both took place in 2016) are used to illustrate this character of metaphorical expressions in sport reports on TV.

Kant against the Animals, Part 2

2018 ◽  
Author(s):  
Christine M. Korsgaard
Keyword(s):  
Intrinsic Value ◽  
Special Kind ◽  
Intrinsic Values ◽  
Limits Of Knowledge

Opponents of Kant suppose he thinks that autonomy gives rational beings a special kind of intrinsic value. Since knowledge of intrinsic values would have to be a kind of metaphysical knowledge, this interpretation is contrary to Kant’s strictures on the limits of knowledge. Rather, Kant thinks that only rational beings can engage in reciprocal lawmaking, which is the source of moral laws. Animals cannot obligate us in the sense of participating in making laws for us. This, however, ignores a second sense in which we can have duties to animals: the laws we make for the treatment of people might also cover the treatment of animals. The chapter ends by explaining why it is hard to get this kind of conclusion using the universalization test.

The Analogical Turn (1884–1887)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Boltzmann Equation ◽  
Statistical Mechanics ◽  
Mechanical System ◽  
Thermal Equilibrium ◽  
Special Kind ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Equipartition Theorem ◽  
Royal Road ◽  
H Theorem

This chapter recounts how Boltzmann reacted to Hermann Helmholtz’s analogy between thermodynamic systems and a special kind of mechanical system (the “monocyclic systems”) by grouping all attempts to relate thermodynamics to mechanics, including the kinetic-molecular analogy, into a family of partial analogies all derivable from what we would now call a microcanonical ensemble. At that time, Boltzmann regarded ensemble-based statistical mechanics as the royal road to the laws of thermal equilibrium (as we now do). In the same period, he returned to the Boltzmann equation and the H theorem in reply to Peter Guthrie Tait’s attack on the equipartition theorem. He also made a non-technical survey of the second law of thermodynamics seen as a law of probability increase.

Suppose and Tell

2020 ◽  
Author(s):  
Timothy Williamson
Keyword(s):  
Conditional Probability ◽  
Natural Deduction ◽  
Special Kind ◽  
Context Sensitivity ◽  
Modal Operator ◽  
Counterfactual Conditionals ◽  
Deduction Rules ◽  
Psychological Sense

The book argues that our use of conditionals is governed by imperfectly reliable heuristics, in the psychological sense of fast and frugal (or quick and dirty) ways of assessing them. The primary heuristic is this: to assess ‘If A, C’, suppose A and on that basis assess C; whatever attitude you take to C conditionally on A (such as acceptance, rejection, or something in between) take unconditionally to ‘If A, C’. This heuristic yields both the equation of the probability of ‘If A, C’ with the conditional probability of C on A and standard natural deduction rules for the conditional. However, these results can be shown to make the heuristic implicitly inconsistent, and so less than fully reliable. There is also a secondary heuristic: pass conditionals freely from one context to another under normal conditions for acceptance of sentences on the basis of memory and testimony. The effect of the secondary heuristic is to undermine interpretations on which ‘if’ introduces a special kind of context-sensitivity. On the interpretation which makes best sense of the two heuristics, ‘if’ is simply the truth-functional conditional. Apparent counterexamples to truth-functionality are artefacts of reliance on the primary heuristic in cases where it is unreliable. The second half of the book concerns counterfactual conditionals, as expressed with ‘if’ and ‘would’. It argues that ‘would’ is an independently meaningful modal operator for contextually restricted necessity: the meaning of counterfactuals is simply that derived compositionally from the meanings of their constituents, including ‘if’ and ‘would’, making them contextually restricted strict conditionals.

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