Related papers: All Concepts are $\mathbb{C}\mathbf{at}^\#$
Given a particular collection of categorical axioms, aimed at capturing properties of the category of locales, we show that if $\mathcal{C}$ is a category that satisfies the axioms then so too is the category $[ G, \mathcal{C}]$ of…
We prove Eilenberg-Watts Theorem for 2-categories of the representation categories $\C\x\Mod$ of finite tensor categories $\C$. For a consequence we obtain that any autoequivalence of $\C\x\Mod$ is given by tensoring with a representative…
We study the category of polynomial functors from finitely generated free groups to a stable infinity-category D. We show that this category is equivalent to the category of excisive functors from pointed animas to D, and also to truncated…
In the well-known settings of category theory enriched in a monoidal category V, the use of V-enriched functor categories and bifunctors demands that V be equipped with a symmetry, braiding, or duoidal structure. In this paper, we establish…
If $\mathcal{C}$ is a cocomplete monoidal category in which tensoring from both sides preserves coequalizers, then the category of monoids over $\mathcal{C}$ is cocomplete. The same holds if $\mathcal{C}$ has regular factorizations and…
In this work, we establish certain enrichments of dual algebraic structures in the setting of monoidal double categories. In more detail, we obtain a tensored and cotensored enrichment of monads in comonads, as well as a tensored and…
Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual…
We specialise a recently introduced notion of generalised dinaturality for functors $T : (\mathcal{C}^\text{op})^p \times \mathcal{C}^q \to \mathcal{D}$ to the case where the domain (resp., codomain) is constant, obtaining notions of ends…
For a category $\mathbb{C}$, a small category $\mathbb{I}$, and a pre-cover relation $\sqsubset$ on $\mathbb C$ we prove, under certain completeness assumptions on $\mathbb C$, that a morphism $g: B\to C$ in the functor category $\mathbb…
The attempt is to give a formal concpet of system, and with this provide a definition of category, that will also satisfy the definition of a system. An axiomatic base is given, for constructing the group of integers. In the process, we…
We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${\mathcal C}$ to be equivalent. This concludes the classification of such module…
We study a number of categorical quasi-uniform structures induced by functors. We depart from a category $\mathcal{C}$ with a proper $(\mathcal{E}, \mathcal{M})$-factorization system, then define the continuity of a $\mathcal{C}$-morphism…
Abbott, Altenkirch, Ghani and others have taught us that many parameterized datatypes (set functors) can be usefully analyzed via container representations in terms of a set of shapes and a set of positions in each shape. This paper builds…
Starting from a comonad G on a category A, and a functor L : B -> A with a right adjoint R : A -> B, we will give a parametrization of the functors K from B to the category of all G-coalgebras that factorize throughout L in terms of…
Category theory provides a collective description of many arrangements in mathematics, such as topological spaces, Banach spaces and game theory. Within this collective description, the perspective from any individual member of the…
We construct a category equivalent to the category $\mathbf{Mon}$ of monoids and monoid homomorphisms, based on categories with strict factorization systems. This equivalence is then extended to the category $\mathbf{Mon_s}$ of unital…
A simple criterion for a functor to be finitary is presented: we call $F$ finitely bounded if for all objects $X$ every finitely generated subobject of $FX$ factorizes through the $F$-image of a finitely generated subobject of $X$. This is…
Whereas formal category theory is classically considered within a $2$-category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of augmented virtual double categories, a notion…
In the first part, we further advance the study of category theory in a strong balanced factorization category C [Pisani, 2008], a finitely complete category endowed with two reciprocally stable factorization systems such that X \to 1 is in…
We study the category of $\mathbf{P}$-equivariant modules over the infinite variable polynomial ring, where $\mathbf{P}$ denotes the subgroup of the infinite general linear group $\mathbf{GL}(\mathbf{C}^\infty)$ consisting of elements…