Tiered Objects

We investigate the foundations of reasoning over infinite data structures by means of set-theoretical structures arising in the sheaf-theoretic semantics of higher-order intuitionistic logic. Our approach focuses on a natural notion of tiering involving an operation of restriction of elements to levels forming a complete Heyting algebra. We relate these tiered objects to final coalgebras and initial algebras of a wide class of endofunctors of the category of sets, and study their order and convergence properties. As a sample application, we derive a general proof principle for tiered objects

On Free Completely Iterative Algebras

For every finitary set functor F we demonstrate that free algebras carry a canonical partial order. In case F is bicontinuous, we prove that the cpo obtained as the conservative completion of the free algebra is the free completely iterative algebra. Moreover, the algebra structure of the latter is the unique continuous extension of the algebra structure of the free algebra. For general finitary functors the free algebra and the free completely iterative algebra are proved to be posets sharing the same conservative completion. And for every recursive equation in the free completely iterative algebra the solution is obtained as the join of an ?-chain of approximate solutions in the free algebra

Contramodules

Contramodules are module-like algebraic structures endowed with infinite summation (or, occasionally, integration) operations satisfying natural axioms. Introduced originally by Eilenberg and Moore in 1965 in the case of coalgebras over commutative rings, contramodules experience a small renaissance now after being all but forgotten for three decades between 1970-2000. Here we present a review of various definitions and results related to contramodules (drawing mostly from our monographs and preprints arXiv:0708.3398, arXiv:0905.2621, arXiv:1202.2697, arXiv:1209.2995, arXiv:1512.08119, arXiv:1710.02230, arXiv:1705.04960, arXiv:1808.00937) - including contramodules over corings, topological associative rings, topological Lie algebras and topological groups, semicontramodules over semialgebras, and a "contra version" of the Bernstein-Gelfand-Gelfand category O. Several underived manifestations of the comodule-contramodule correspondence phenomenon are discussed.Comment: LaTeX 2e with pb-diagram and xy-pic; 93 pages, 3 commutative diagrams; v.4: updated to account for the development of the theory over the four years since Spring 2015: introduction updated, references added, Remark 2.2 inserted, Section 3.3 rewritten, Sections 3.7-3.8 adde

On Terminal Coalgebras Derived from Initial Algebras

A number of important set functors have countable initial algebras, but terminal coalgebras are uncountable or even non-existent. We prove that the countable cardinality is an anomaly: every set functor with an initial algebra of a finite or uncountable regular cardinality has a terminal coalgebra of the same cardinality. We also present a number of categories that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. Finally, for finitary set functors we prove that the initial algebra mu F and terminal coalgebra nu F carry a canonical ultrametric with the joint Cauchy completion. And the algebra structure of mu F determines, by extending its inverse continuously, the coalgebra structure of nu F

A tale of three homotopies

For a Koszul operad $\mathcal{P}$, there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy $\mathcal{P}$-algebras. Some of those approaches are known to give rise to the same notions. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette.Comment: 22 pages, final versio

Canonical extension and canonicity via DCPO presentations

The canonical extension of a lattice is in an essential way a two-sided completion. Domain theory, on the contrary, is primarily concerned with one-sided completeness. In this paper, we show two things. Firstly, that the canonical extension of a lattice can be given an asymmetric description in two stages: a free co-directed meet completion, followed by a completion by \emph{selected} directed joins. Secondly, we show that the general techniques for dcpo presentations of dcpo algebras used in the second stage of the construction immediately give us the well-known canonicity result for bounded lattices with operators.Comment: 17 pages. Definition 5 was revised slightly, without changing any of the result