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Register a new userOn monoidal equivalences and Ann-equivalences
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In this paper, we show another proof of the problem by constructing a strict monoidal category M(C) consisting of M-functors and M-morphisms of a category C and we prove C is equivalent to it. The proof is based on a basic character of monoidal equivalences. Ideas and techniques of these proofs can been used to prove the equivalence between an Ann-category and an almost strict Ann-category.
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It is well known that a resolving subcategory $mathcal{A}$ of an abelian subcategory $mathcal{E}$ induces several derived equivalences: a triangle equivalence $mathbf{D}^-(mathcal{A})to mathbf{D}^-(mathcal{E})$ exists in general and furthermore restricts to a triangle equivalence $mathbf{D}^{mathsf{b}}(mathcal{A})to mathbf{D}^{mathsf{b}}(mathcal{E})$ if $operatorname{res.dim}_{mathcal{A}}(E)<infty$ for any object $Ein mathcal{E}$. If the category $mathcal{E}$ is uniformly bounded, i.e. $operatorname{res.dim}_{mathcal{A}}(mathcal{E})<infty$, one obtains a triangle equivalence $mathbf{D}(mathcal{A})to mathbf{D}(mathcal{E})$. In this paper, we show that all of the above statements hold for preresolving subcategories of (one-sided) exact categories. By passing to a one-sided language, one can remove the assumption that $mathcal{A}subseteq mathcal{E}$ is extension-closed completely from the classical setting, yielding easier criteria and more examples. To illustrate this point, we consider the Isbell category $mathcal{I}$ and show that $mathcal{I}subseteq mathsf{Ab}$ is preresolving but $mathcal{I}$ cannot be realized as an extension-closed subcategory of an exact category. We also consider a criterion given by Keller to produce derived equivalences of fully exact subcategories. We show that this criterion fits into the framework of preresolving subcategories by considering the relative weak idempotent completion of said subcategory.
We give a characterisation of functors whose induced functor on the level of localisations is an equivalence and where the isomorphism inverse is induced by some kind of replacements such as projective resolutions or cofibrant replacements.
We apply the Auslander-Buchweitz approximation theory to show that the Iyama and Yoshinos subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitzs triangle equivalence from Iwanaga-Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitzs triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga-Gorenstein rings and Gorenstein algebras
For models of concurrent and distributed systems, it is important and also challenging to establish correctness in terms of safety and/or liveness properties. Theories of distributed systems consider equivalences fundamental, since they (1) preserve desirable correctness characteristics and (2) often allow for component substitution making compositional reasoning feasible. Modeling distributed systems often requires abstraction utilizing nondeterminism which induces unintended behaviors in terms of infinite executions with one nondeterministic choice being recurrently resolved, each time neglecting a single alternative. These situations are considered unrealistic or highly improbable. Fairness assumptions are commonly used to filter system behaviors, thereby distinguishing between realistic and unrealistic executions. This allows for key arguments in correctness proofs of distributed systems, which would not be possible otherwise. Our contribution is an equivalence spectrum in which fairness assumptions are preserved. The identified equivalences allow for (compositional) reasoning about correctness incorporating fairness assumptions.
In this paper, we have studied the axiomatics of {it Ann-categories} and {it categorical rings.} These are the categories with distributivity constraints whose axiomatics are similar with those of ring structures. The main result we have achieved is proving the independence of the axiomatics of Ann-category definition. And then we have proved that after adding an axiom into the definition of categorical rings, we obtain the new axiomatics which is equivalent to the one of Ann-categories.
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