category-theory

symmetric monoidal (∞,1)-category of spectra The notion of coring is a generalization of a -coalgebra. While for a coalgebra must be a commutative ring (often a field), a coring is defined over a general noncommutative ring or even an associative algebra . Whereas a coalgebra structure is defined on a -module (if is a field, it is a vector space) – which may be regarded as a central -bimodule – a…

Profunctor Equipment in Haskell Posted by Bartosz Milewski under Category Theory,

abstract duality: opposite category, concrete duality: dual object, dualizable object, fully dualizable object, dualizing object Examples between higher geometry/higher algebra Langlands duality, geometric Langlands duality, quantum geometric Langlands duality In QFT and String theory higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-topose…

Formal -category theory is to -category theory what formal category theory is to 1-category theory, that is: a synthetic approach to -categories standing in relation to synthetic homotopy theory as -category theory stands to homotopy theory. Consequently, formal -category theory has also been called synthetic -category theory. There have been two main styles of approaches to formal -category theo…

Anders Kock is a mathematician at Aarhus University, Denmark. He works on category theory/topos theory and particularly on synthetic differential geometry. On the 2-category of complete categories being monadic over Cat: Introducing the notion of strong monads and relating to monoidal monads and commutative monads: Anders Kock, Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970)…

This is the problem in question from Mac Lane's "Categories for the Working Mathematician", chapter 2, section 4, exercise 1: For $R$ a ring, describe $R\mbox{-}\mathbf{Mod}$ as a full ...

Given an object in a category the domain functor from the slice category to is a fibered category (i.e. Grothendieck fibration). Any fibered category equivalent to the is said to be representable. This is because under the Grothendieck construction representable fibered categories correspond precisely to representable functors : the category is the category of elements of the representable functo…

With braiding With duals for objects category with duals (list of them) dualizable object (what they have) ribbon category, a.k.a. tortile category With duals for morphisms With traces Closed structure Special sorts of products Semisimplicity Morphisms Internal monoids Examples Theorems In higher category theory The formalism of Markov categories and copy-discard categories is a categorical appro…

Recall the following familiar 1-categorical statement: The idea of -toposes is to generalize the above situation from to (recall the notion of (n,r)-category and see the general discussion at ∞-topos): Recall that sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes and that the inclusion functor is necessarily an accessible functor. This characterization has…

André Joyal is a Canadian mathematician, a professor at Université du Québec à Montréal. He got his PhD in 1971 from Université de Montréal. His wide mathematical work, is mainly in category theory, topos theory and abstract homotopy theory. His works include a wide generalization of Galois theory with Myles Tierney, the combinatorial ideas of “Joyal’s species”, discovery of the category structur…

Given a site , then a local epimorphism is a morphism in the category of presheaves over the site which becomes an epimorphism under sheafification. More abstractly, for a small category, one says axiomatically that a system of local epimorphisms is a system of morphisms in the presheaf category that has the closure properties expected of epimorphisms under composition and under pullback. There i…

An algebraic pattern is a blueprint for a notion of functors on a fixed category satisfying a Segal condition, suitable for formalizing homotopy-coherent algebra in the Cartesian setting. An algebraic pattern is an (∞,1)-category together with the following data: a pair of wide subcategories , whose morphisms are called inert and active morphisms, and a full subcategory , whose objects are called…

Definitions Transfors between 2-categories Morphisms in 2-categories Structures in 2-categories Limits in 2-categories Structures on 2-categories A double category is an internal category in Cat. Similarly, a double groupoid is an internal groupoid in Grpd. However, these definitions obscure the essential symmetry of the concepts. We think of a double category as having We may picture a 2-cell in…

Given an object $c$ of a category $\mathsf C$, we say that: $c$ is small when $\mathrm{Hom}_{\mathsf C}(c,-)$ commutes with $\kappa$-filtered colimits for some cardinal $\kappa$. $c$ is compact when ...

Yesterday a friend and I had a conversation about category theory, how it can be a useful pattern description language, but also about how people have unrealistic expectations for it, believing category theory can deliver something for nothing. Later I ran across the following post from Qiaochu Yuan. It felt as if he had overheard […] The post The mythology of category theory first appeared on Jo…

Koenig's theorem says that, given families of sets $(A_i)_{i \in I}$, $(B_i)_{i \in I}$, if there exists an embedding $\prod_{i \in I} B_i \hookrightarrow \coprod_{i \in I} A_i$, then for some $i \in ...

Let $C$ be a small category with finite limits. Let $D$ have all small limits and colimits. Any functor $F : C \to D$ corresponds to a functor $\hat{F} : \mathbf{Sets}^{C^{\mathrm{op}}} \to D$ ...

Let be a differential graded category. A twisted complex in is a graded set of objects of , such that only finitely many are not the zero object; a set of morphisms such that ; . The differential graded category of twisted complexes in has as objects twisted complexes and with differential given on given by The construction of categories of twisted complexes is functorial in that for a dg-functor…

With braiding With duals for objects category with duals (list of them) dualizable object (what they have) ribbon category, a.k.a. tortile category With duals for morphisms With traces Closed structure Special sorts of products Semisimplicity Morphisms Internal monoids Examples Theorems In higher category theory A cartesian closed category (sometimes: ccc) is a category with finite products which…

A Freyd category is one way to axiomatize models of call-by-value? programming languages. It abstracts the structure of the Kleisli category of a monad, consisting of a category that models values and another category with the same objects that models computations. Freyd categories thus provide a categorical semantics for typed programming languages with side effects such as memory access or prin…