Jirí Rosický

On combinatorial model categories

Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly generated. He has not published his results yet but proofs of some of them were presented by T. Beke, D. Dugger or J. Lurie. We are contributing to this endeavour by some new results about homotopy equivalences, weak equivalences and cofibrations in combinatorial model categories.

Accessible Categories, Saturation and Categoricity

Model theoretic concepts of saturation and categoricity are studied in the context of accessible categories.

Abstract elementary classes and accessible categories

Abstract We investigate properties of accessible categories with directed colimits and their relationship with categories arising from Shelahʼs Abstract Elementary Classes. We also investigate ranks of objects in accessible categories, and the effect of accessible functors on ranks.

Class-locally presentable and class-accessible categories

We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the abstract homotopy theory.

More on directed colimits of models

M. Richter has proved that whenever a classK of ∑-structures has a finitary first-order axiomatization then the inclusionK ↪Str ∑ preserves all existing directed colimits (see [7]). We will generalize this result to classes of ∑-structures having an infinitary first-order axiomatization in a larger signature ∑′. We will also show that, as categories, these classes have a natural characterization.

Classification theory for accessible categories

We show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht-Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.

On abstract data types presented by multiequations

Equational presentation of abstract data types is generalized to presentation by multiequations, i.e., exclusive-ors of equations, in order to capture parametric data types such as array or set. Multiinitial-algebra semantics for such data types is introduced. Classes of algebras described by multiequations are characterized.

METRIC ABSTRACT ELEMENTARY CLASSES AS ACCESSIBLE CATEGORIES

We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed colimits, with concrete

On geometric and finitary sketches

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Erratum to “Left-determined model categories and universal homotopy theories”

-directed colimits and concrete monomorphisms. More broadly, we define a notion of