Related papers: The existential completion
We define the notion of exact completion with respect to an existential elementary doctrine. We observe that the forgetful functor from the 2-category exact categories to existential elementary doctrines has a left biadjoint that can be…
We provide a thorough algebraic analysis of three known completions having a central role in the exact completions of Lawvere's doctrines: the one adding comprehensive diagonals (i.e. forcing equality on terms to coincide with the equality…
This paper aims to apply the tool of generalized existential completions of conjunctive doctrines, concerning a class $\Lambda$ of morphisms of their base category, to deepen the study of regular and exact completions of existential…
We extend the notion of exact completion on a weakly lex category to elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which freely adds effective quotients and extensional equality. We note that…
We study the relationship between cartesian bicategories and a specialisation of Lawvere's hyperdoctrines, namely elementary existential doctrines. Both provide different ways of abstracting the structural properties of logical systems: the…
Recently, Abbadini and Guffanti gave an algebraic proof of Herbrand's theorem using a completion for Lawvere doctrines that freely adds existential and universal quantifiers. A more direct argument can be given by only completing with…
This thesis focuses on topics in 2-category theory: in particular on double categories, pseudomonads and codescent objects. In Chapter 2 we recall all the necessary notions. In Chapter 3 we show that factorization systems can be…
We present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic.
A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. For all finitary (and, more generally, all precontinuous) set functors the initial…
A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. For all finitary (and, more generally, all precontinuous) set functors the initial…
We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg-Moore algebras for an oplax monoidal monad, we always have a natural…
In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form $\forall x_0 \exists x_1 \dots \exists x_n \bigwedge x_i R_\lambda x_j$. We prove that many properties of these logics, such…
In this work, we fill the gap between the elementary quotient completion introduced by Maietti and Rosolini and the exact completion of a category with weak finite limits, as described by Carboni and Vitale. To achieve this, we generalize…
Idempotent analogues of convexity are introduced. It is proved that the category of algebras for the capacity monad in the category of compacta is isomorphic to the category of $(\max,\min)$-idempotent biconvex compacta and their biaffine…
We categorify cocompleteness results of monad theory, in the context of pseudomonads. We first prove a general result establishing that, in any 2-category, weighted bicolimits can be constructed from oplax bicolimits and bicoequalizers of…
We show that the theory of De Morgan algebras has a model completion and axiomatise it. Then we prove that it is $\aleph_0$-categorical and describe definable and algebraic closures in that theory. We also obtain similar results for…
We use Kan injectivity to axiomatise concepts in the 2-category of topoi. We showcase the expressivity of this language through many examples, and we establish some aspects of the formal theory of Kan extension in this 2-category (pointwise…
The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…
For representation by partial functions in the signature with intersection, composition and antidomain, we show that a representation is meet complete if and only if it is join complete. We show that a representation is complete if and only…
We generalize the correspondence between theories and monads with arities of arXiv:1101.3064 to $\infty$-categories. Additionally, we introduce the notion of complete theories that is unique to the $\infty$-categorical case and provide a…