Related papers: Linearly Distributive Fox Theorem
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a…
We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central…
Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation $\mathsf{K}$, which all…
Let $\mathcal C$ be a category with finite colimits, and let $(\mathcal E,\mathcal M)$ be a factorisation system on $\mathcal C$ with $\mathcal M$ stable under pushouts. Writing $\mathcal C;\mathcal M^{\mathrm{op}}$ for the symmetric…
In this note, we explain in some detail how one can fiberwise localize a (co)lax symmetric monoidal infinity-category. This construction was tacitly used in Section 5 of our recent paper "On the equivalence of the Lurie's infinity-operads…
Linearity and dependency analyses are key to several applications in computer science, especially, in resource management and information flow control. What connects these analyses is that both of them need to model at least two different…
The skew monoidal categories of Szlach\'anyi are a weakening of monoidal categories where the three structural laws of left and right unitality and associativity are not required to be isomorphisms but merely transformations in a particular…
We verify that Kelly's constructions of the internal Hom for enriched categories extends naturally to lax functors taking their values in a symmetric monoidal category. Our motivation is to set up a `calculus on lax functors' that will host…
We upgrade the classical operation of \textit{isomonodromic deformations} along a path $\gamma$ to a functor $\mathbb{P}_{\gamma}$ between categories of flat connections with logarithmic singularities along a divisor $D$, which itself…
Liftings of endofunctors on sets to endofunctors on relations are commonly used to capture bisimulation of coalgebras. Lax versions have been used in those cases where strict lifting fails to capture bisimilarity, as well as in modeling…
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…
Categories can be identified -- up to isomorphism -- with polynomial comonads on Set. The left Kan extension of a functor along itself is always a comonad -- called the density comonad -- so it defines a category when its carrier is…
We develop layered monoidal theories -- a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams, monoidal theories provide a graphical…
In categories of linear relations between finite dimensional vector spaces, composition is well-behaved only at pairs of relations satisfying transversality and monicity conditions. A construction of Wehrheim and Woodward makes it possible…
This article gives a solid theoretical grounding to the observation that cubical structures arise naturally when working with parametricity. We claim that cubical models are cofreely parametric. We use categories, lex categories or clans as…
Actions of monoidal categories on categories, also known as actegories, have been familiar to category theorists for a long time, and yet a comprehensive overview of this topic seems to be missing from the literature. Recently, actegories…
We provide a calculus of mates for functors to the $\infty$-category of $\infty$-categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do…
We extend to general Cartesian categories the idea of Coherent Differentiation recently introduced by Ehrhard in the setting of categorical models of Linear Logic. The first ingredient is a summability structure which induces a partial…
Causal deep learning (CDL) is a new and important research area in the larger field of machine learning. With CDL, researchers aim to structure and encode causal knowledge in the extremely flexible representation space of deep learning…
Grothendieck's theory of fibred categories establishes an equivalence between fibred categories and pseudo functors. It plays a major role in algebraic geometry and categorical logic. This paper aims to show that fibrations are also very…