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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…
We define notions of regularity and (Barr-)exactness for 2-categories. In fact, we define three notions of regularity and exactness, each based on one of the three canonical ways of factorising a functor in Cat: as (surjective on objects,…
For a small quantaloid $\mathcal{Q}$, we introduce $\mathcal{M}$-(co)complete $\mathcal{Q}$-categories, i.e., (co)complete $\mathcal{Q}$-categories up to Morita equivalence, as Eilenberg--Moore algebras of the presheaf monad on the category…
We provide a criterion for the existence of right approximations in cocomplete additive categories; it is a straightforward generalisation of a result due to El Bashir. This criterion is used to construct adjoint functors in homotopy…
Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme. We demonstrate these similarities by showing how this simple scheme encompasses…
In 2017, Bauer, Johnson, Osborne, Riehl, and Tebbe (BJORT) showed that the Abelian functor calculus provides an example of a Cartesian differential category. The definition of a Cartesian differential category is based on a differential…
A new hierarchy of "exact" unification types is introduced, motivated by the study of admissibility for equational classes and non-classical logics. In this setting, unifiers of identities in an equational class are preordered, not by…
Many economic theory models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. We provide a principled framework for scaling results from such models by removing these finiteness…
We prove a general theorem which includes most notions of "exact completion". The theorem is that "k-ary exact categories" are a reflective sub-2-category of "k-ary sites", for any regular cardinal k. A k-ary exact category is an exact…
We consider the canonical pseudodistributive law between various free limit completion pseudomonads and the free coproduct completion pseudomonad. When the class of limits includes pullbacks, we show that this consideration leads to notions…
A new hierarchy of "exact" unification types is introduced, motivated by the study of admissible rules for equational classes and non-classical logics. In this setting, unifiers of identities in an equational class are preordered, not by…
We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and…
Motivated by the classical correspondence between short exact sequences and splitting properties in module theory, this paper examines the projective and injective analogues within the category of Lie algebras. We first show that no Lie…
We develop a theory of completeness for weight structures on stable categories, dual to the theory of complete t-structures. As in the bounded case, we show that complete weight structures are determined by their weight heart, giving rise…
In an abelian category $\mathscr{A}$ with small ${\rm Ext}$ groups, we show that there exists a one-to-one correspondence between any two of the following: balanced pairs, subfunctors $\mathcal{F}$ of ${\rm Ext}^{1}_{\mathscr{A}}(-,-)$ such…
This article provides a general framework in the context of category theory where one can recognize as particular instances of the same abstract construction several notions of completion, envelope, and hull, such as the Boolean algebra…
Projectivity and injectivity are fundamental notions in category theory. We consider natural weakenings termed semiprojectivity and semiinjectivity, and study these concepts in different categories. For example, in the category of metric…
Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every…
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 $\aleph_1$-directed colimits and concrete monomorphisms. More…
We use type-theoretic techniques to present an algebraic theory of $\infty$-categories with strict units. Starting with a known type-theoretic presentation of fully weak $\infty$-categories, in which terms denote valid operations, we extend…