Related papers: A Generalised Exactness Structure for Sets
We apply the notion of relative adjoint functor to generalise closed monoidal categories. We define representations in such categories and give their relation with left actions of monoids. The translation of these representations under lax…
Rough sets are approximations of concrete sets. The theory of rough sets has been used widely for data-mining. While it is well-known that adjunctions are underlying in rough approximations, such adjunctions are not enough for…
This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural…
We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category…
Consider a reflection from a finitely-complete category $\mathbb{C}$ into its full subcategory $\mathbb{M}$, with unit $\eta :1_\mathbb{C}\rightarrow HI$. Suppose there is a left-exact functor $U$ into the category of sets, such that $UH$…
The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting set-theoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional…
Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $\infty$-categories. One of our main results is an…
Many mathematical objects can be represented as functors from finitely-presented categories $\mathsf{C}$ to $\mathsf{Set}$. For instance, graphs are functors to $\mathsf{Set}$ from the category with two parallel arrows. Such functors are…
Given a pair of adjoint functors between two arbitrary categories it induces mutually inverse equivalences between the full subcategories of the initial ones, consisting of objects for which the arrows of adjunction are isomorphisms. We…
Diers developed a general theory of right multi-adjoint functors leading to a purely categorical, point-set construction of spectra. Situations of multiversal properties return sets of canonical solutions rather than a unique one. In the…
We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the…
The general Galois theory for functions and relational constraints over arbitrary sets described in the authors' previous paper is refined by imposing algebraic conditions on relations.
There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms…
The concept of Galois connections (i.e., adjoint pairs between posets) is ubiquitous in mathematics. In representation theory, it is interesting because it naturally induces the adjoint quadruple between the categories of persistence…
For a given family $\{(\mathrm{q}_i, \mathrm{t}_i, \mathrm{p_i} )\}_{i \in I}$ of adjoint triples between exact categories $\mathcal{C}$ or $\mathcal{D}$, we show that any cotorsion pair in $\mathcal{C}$ and $\mathcal{D}$ yield two…
We prove an essentially surjective Galois-correspondence-like functor for $n$-stacks. More specifically, it gives an essentially surjective functor from the $\infty$-category of $n$-stacks of finite sets with an action of the fundamental…
This paper introduces the notion of complete connectedness of a Grothendieck topos, defined as the existence of a left adjoint to a left adjoint to a left adjoint to the global sections functor, and provides many examples. Typical examples…
Given a right adjoint functor between triangulated categories and an object in the target category, we show that the unit map of adjunction on that object is a split monomorphism if and only if the object belongs to the additive closure of…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
In this work, we investigate an effective method for showing that functors between categories are left adjoints. The method applies to a large class of categories, namely locally finitely presentable categories, which are ubiquitous in…