Applied Categorical Structures (2021) 29:235–248 https://doi.org/10.1007/s10485-020-09614-w Codenseness and Openness with Respect to an Interior Operator Fikreyohans Solomon Assfaw 1 · David Holgate 1 Received: 13 September 2019 / Accepted: 9 October 2020 / Published online: 26 October 2020 © Springer Nature B.V. 2020 Abstract Working in an arbitrary category endowed with a fixed (E , M)-factorization system such that M is a fixed class of monomorphisms, we first define and study a concept of codense morphisms with respect to a given categorical interior operator i . Some basic properties of these morphisms are discussed. In particular, it is shown that i -codenseness is preserved under both images and dual images under morphisms in M and E , respectively. We then introduce and investigate a notion of quasi-open morphisms with respect to i . Notably, we obtain a characterization of quasi i -open morphisms in terms of i -codense subobjects. Fur- thermore, we prove that these morphisms are a generalization of the i -open morphisms that are introduced by Castellini. We show that every morphism which is both i-codense and quasi i-open is actually i-open. Examples in topology and algebra are also provided. Keywords Interior operator · Codenseness · Openness · Quasi-openness Mathematics Subject Classification 06A15 · 18A20 · 54B30 1 Introduction A categorical closure operator on an arbitrary category is a family of functions (on suitably defined subobject lattices) which are expansive, order preserving and compatible with tak- ing images or equivalently, preimages, in the same way as the usual topological closure is compatible with continuous maps. The formal theory of categorical closure operators was introduced by Dikranjan and Giuli [11] and then developed by these authors and Tholen [12]. The theory was largely inspired by Salbany’s paper [23], where regular closure operators on the category of topological spaces and continuous maps were introduced. These operators Communicated by Maria Manuel Clementino. B Fikreyohans Solomon Assfaw fikreyohans@aims.ac.za David Holgate dholgate@uwc.ac.za 1 Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville 7535, South Africa 123