Related papers: Locally Cartesian Closed Categories
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…
The present paper gives a generalization of cartesian closed categories, called cartesian closed categories with dependence, whose strict version induces categories with families that support 1-, Sigma- and Pi-types in the strict sense.…
A folklore result in category theory is that a (weakly) Cartesian closed category with finite co-products is distributive. Usually, the proof of this small result is carried on using the fact that the exponential functor is right adjoint to…
We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to…
We are checking the closed categories beginning with the category of sets and ending with the category of categories. The novelty is a generalizing the notion of adjoint functors to the joint pair of functors in the category of directed…
The cartesian structure possessed by relations, spans, profunctors, and other such morphisms is elegantly expressed by universal properties in double categories. Though cartesian double categories were inspired in part by the older program…
We establish a DK-equivalence between the relative category of $\pi$-tribes and the relative category of locally cartesian closed quasicategories. From this follows one of the internal languages conjecture: Martin-L\"of type theory with…
Graduated locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and Boolean algebras. A finitary functor between graduated locally finitely presentable categories is…
A theorem is derived which (i) provides a new class of subfactors which may be interpreted as generalized asymptotic subfactors, and which (ii) ensures the existence of two-dimensional local quantum field theories associated with certain…
As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is…
There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the hom-enriched context. Then we do it in the cocompletion of a bicategory with respect to…
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…
The definition of the local fractional derivative has been generalised to the orders beyond the critical order. This makes it possible to retain more terms in the local fractional Taylor expansion leading to better approximation. This also…
We present a categorical model for intuitionistic linear logic where objects are polynomial diagrams and morphisms are simulation diagrams. The multiplicative structure (tensor product and its adjoint) can be defined in any locally…
We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.
We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over…
We show that the category of categories with pullbacks and pullback preserving functors is cartesian closed.
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…
Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…
We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external…