In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, bi-interpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We provide some examples to separate notions across categories. In contrast, we show that, in some c...

We categorify the R-matrix isomorphism between tensor products of minuscule representations of Uq(sln) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of Uq(sl2) which are related to representations of Uq(sln) by quant...

The problem of iconicity has been a recurring theme in linguistic inquiry into the nature of the linguistic sign. The property of iconicity includes not only the narrow category of sound symbolism, but also the broad categories of isomorphism and metaphor. This paper surveys recent research, and offers a classification of diagrammatic and metaphorical iconicity in language. Problems of structur...

We categorify the R-matrix isomorphism between tensor products of minuscule representations of Uq(sln) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of Uq(sl2) which are related to representations of Uq(sln) by quant...

From an arbitrary temporal logic institution we show how to set-up the corresponding institution of objects. The main properties of the resulting institution are studied and used in establishing the denotational, categorial semantics of several basic object specification constructs, namely aggregation (parallel composition), interconnection, abstraction (interfacing) and monotonic specializatio...

Automata over a symmetric monoidal category M are introduced, and a multi-step simulation is defined among such automata. The collection of M automata is given the structure of a 2-category on the same objects as M , in which the vertical structure is determined by groups of indistinguishable simulations. TwoM -automata are called simulation equivalent if they are connected by an isomorphism of...

In this paper, we introduce and study notions like state-\linebreak distinguishability, input-distinguishability and output completeness of states of a crisp deterministic fuzzy automaton. We show that for each crisp deterministic fuzzy automaton there corresponds a unique (up to isomorphism), equivalent distinguished crisp deterministic fuzzy automaton. Finally, we introduce two axioms related...

The category is introduced as an ordered 5-tuple of the form 〈O,M,dom,cod, ·, id〉 where O (objects) and M (morphisms) are arbitrary nonempty sets, dom and cod map M onto O and assign to a morphism domain and codomain, · is a partial binary map from M×M to M (composition of morphisms), id applied to an object yields the identity morphism. We define the basic notions of the category theory such a...

Exercises 1.2 (a) Composition in RC is well-defined by representatives. (b) A morphism of RC is invertible ⇐⇒ it is a class of strong equivalences ⇐⇒ its counit is an isomorphism. (c) Objects of RC are isomorphic iff they are equivalent categories. (d) RC/T ' Copt(T ) (?). Rigid comparisons embody an important idea from domain theory: approximation. This must have the property that if X ′ appro...