category-theory

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…

hom-set, hom-object, internal hom, exponential object, derived hom-space loop space object, free loop space object, derived loop space Given categories and , the functor category – written or – is the category whose morphisms are natural transformations between these functors. Discussion in homotopy type theory. Note: the HoTT book calls a internal category in HoTT a “precategory” and a univalent…

Marcy Robertson is a lecturer at the University of Melbourne. She works on operad theory, homotopy theory, and higher category theory. On an (∞,1)-operad version of modular operads:

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 In vertical categorification of how monoids/monoid objects may act on other objects (…

A cocone under a diagram is an object equipped with morphisms from each vertex of the diagram into it, such that all new diagrams arising this way commute. A cocone which is universal is a colimit. The dual notion is cone . Let and be categories; we generally assume that is small. Let be a functor (called a diagram in this situation). Then a cocone (or inductive cone) over is a pair of an object …

The Bunge Family: Mario, Eric, Marta and Silvia Some mathematical projects are less about proving theorems and more about bringing people together. Editing the special volume of Theory and Applications of Categories in honor of Marta Bunge felt very much like that. I worked on it together with Maria Manuel Clementino and Jonathan Funk , and what stayed with me throughout was not just the range of…

model category, model -category Definitions Morphisms Universal constructions Refinements Producing new model structures Presentation of -categories Model structures for -groupoids on chain complexes/model structure on cosimplicial abelian groups related by the Dold-Kan correspondence for equivariant -groupoids for rational -groupoids for rational equivariant -groupoids for -groupoids for -groups…

nLab Model Categories Context Model category theory model category, model -category Definitions Morphisms Universal constructions Refinements Producing new model structures Presentation of -categories Model structures for -groupoids for ∞-groupoids for equivariant -groupoids for rational -groupoids for rational equivariant -groupoids for -groupoids for -groups for -algebras general -algebras spec…

Victor Ostrik is a mathematicain at University of Oregon. He has been one of the main figures in the development of the theory of fusion categories. On module categories, weak Hopf algebras and modular invariants: Pavel Etingof, Dmitri Nikshych, Victor Ostrik, On fusion categories, Annals of Mathematics Second Series, Vol. 162, No. 2 (Sep., 2005), pp. 581-642 (arXiv:math/0203060, jstor:20159926) …

Let be a pair of adjoint functors (an adjunction in Cat). The envelope of the adjunction, denoted , is the category whose objects are quadruples such that and are each other’s mate, and whose morphisms are pairs such that or equivalently . A functor is left adjoint to a functor if and only if there is an isomorphism of comma categories and this isomorphism commutes with the forgetful functors to …

(The special case in Cat of the general notion of adjunction.) The concept of adjoint functors [Kan (1958)] is a key concept in category theory — if not the key concept — and it is in large part through the manifold identification of examples of adjoint functors appearing ubiquitously in the practice of mathematics that category theoretic tools are brought to use in general mathematics. Abstractl…

homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Definitions Paths and cylinders Homotopy groups Basic facts Theorems Homotopy products are Cartesian products in homotopy theory, hence are a special case of homotop…

Homotopy coproducts are a special case of homotopy colimits, when the indexing diagram is a discrete category. Homotopy coproducts can be defined in any relative category, just like homotopy colimits, but practical computations are typically carried out in presence of additional structures such as model structures. In any model category, the homotopy coproduct of a family of objects can be comput…

A locally cartesian closed functor is a functor between locally cartesian closed categories that preserves the locally cartesian closed structure up to coherent isomorphism. cartesian closed category, locally cartesian closed category cartesian closed functor, locally cartesian closed functor cartesian closed model category, locally cartesian closed model category cartesian closed (∞,1)-category,…

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 symmetric monoidal closed category is which as such is also: For monads on symmetri…

This paper develops a formal categorical analysis of distinction collapse and its consequences for invariant reconstruction. We consider representation functors between categories and show that when such functors collapse distinctions—either by failing to be faithful or by identifying non-isomorphic objects—no reconstruction functor can preserve distinctions up to natural isomorphism. In such cas…