In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. Two counterexamples are provided for that goal, and detailed proofs are given. In par...

We consider derived invariants of varieties in positive characteristic arising from topological Hochschild homology. Using theory developed by Ekedahl and Illusie-Raynaud their study the slope spectral sequence, we examine behavior under equivalences various $p$-adic quantities related to Hodge-Witt crystalline cohomology groups, including numbers, domino numbers. As a consequence, obtain restr...

Two logics are implicationally equivalent if the axioms and inference rules of each imply the axioms of the other. Characterizing the implicational equivalences of various formulations of the sentential calculi is foundational to the study of logic. Using an automated deduction system, I show that Łukasiewicz's CN sentential calculus can be derived from Church's P2; the proof appears to be novel.

We present a novel notion of stable objects in a triangulated category. This Postnikov-stability is preserved by equivalences. We show that for the derived category of a projective variety this notion includes the case of semistable sheaves. As one application we compactify a moduli space of stable bundles using genuine complexes via Fourier-Mukai transforms. MSC 2000: 14F05, 14J60, 14D20

In this note we show that two nite dimensional algebras have the same representation type if they are stably equivalent of Morita type. Stable equivalences of Morita type were introduced for blocks of group algebras by Brou e 2], see also 7]. The concept is motivated by a result of Rickard. In 9], he proved that any derived equivalence between nite dimensional self-injective algebras and ? indu...

In an earlier paper the notion of a filtered derived equivalence was introduced, and it was shown that if two K3 surfaces admit such an equivalence then they are isomorphic. In this paper we study more refined aspects of filtered derived equivalences related to the action on the cohomological realizations of the Mukai motive. It is shown that if a filtered derived equivalence between K3 surface...

In this paper a set of Petri net equivalences is proposed. A correlation between all introduced equivalences is established, and a lattice of implications is obtained. In addition equivalences are examined on sequential nets. The process equivalences are demonstrated to be well discerning on this net class.