category-theory

A reflective subcategory is a full subcategory such that objects and morphisms in have “reflections” and in . Every object in looks at its own reflection via a morphism and the reflection of an object is equipped with an isomorphism . A canonical example is the inclusion of the category of abelian groups into the category of groups, whose reflector is the operation of abelianization. A useful pro…

A semifunctor is a homomorphism between semicategories, like a functor is a homomorphism between categories. A semifunctor from a semicategory to a semicategory is a map sending each object to an object and each morphism in to morphism in , such that If is a category, then need not preserve its identity morphisms, but the composition axiom does require that it send them to idempotents in . In Rel…

homotopy hypothesis-theorem delooping hypothesis-theorem stabilization hypothesis-theorem Opetopic type theory (Finster 12) is a higher dimensional directed homotopy type theory for omega-categories, i.e. of infinity-categories in the full sense of -categories. Specifically, it realizes the higher-dimensional horn-filler conditions in the definition of opetopic omega-categories due to Palm as inf…

Foliated categories (French: catégories feuilletées), or simply foliations (not to be confused with the notion of foliations in differential geometry), were introduced by Jean Bénabou in unpublished work dating back to 1984. They are a weaker structure than fibered categories, but still allow one to test for various standard properties of functors fibre-wise. A functor makes its domain category a…

Andrée Charles Ehresmann (née Bastiani) is a category theorist. She is editor in chief of the journal Cahiers de Topologie et Géométrie Différentielle Catégoriques, founded by her late husband Charles Ehresmann. Andrée Ehresmann’s early papers were published under her maiden name Andrée Bastiani, and during a transition period under the hyphenated surname Bastiani-Ehresmann. On internalization of…

The notion of minimal objects generalizes the notion of minimal elements of preorders. Intuitively, minimal elements are ones that don’t have anything below them, even though they might not be the minimum. A ‘minimum object’ would be instead an initial object. Minimal objects have been first defined in (Adámek et al. ‘12) in the context of coalgebra theory, relative to an orthogonal factorization…

Suppose we have a category $C$ of "structured sets" in the usual sense. The goal is to compute universal objects in this category $C$ as sets with their respective structures. To elaborate, ...

Riehl and Verity proved in [RV16] that the strict 2-category of the walking adjunction, usually denoted $\mathrm{Adj}$, has the expected universal property as the homotopy coherent walking adjunction. ...

An automorphism of an object in a category is an isomorphism . In other words, an automorphism is an endomorphism that is an isomorphism. Given an object , the automorphisms of form a group under composition, the automorphism group of , which is a submonoid of the endomorphism monoid of : which may be written if the category is understood. Up to equivalence, every group is an automorphism group; …

The theory of locally presentable categories, accessible categories and accessible functors is a powerful theory that includes many large categories and functors of interest. This theory often gives ...

A metric space $(X,d)$ can be viewed as a category enriched over the non-negative reals (with $\infty$), where the objects are points and the hom-objects are distances. Functors correspond to ...

I posted a question yesterday and it was closed due to duplicate because some thought I was asking for a definition of canonical morphism. I would make it a more focused question: In category thory I ...

homotopy hypothesis-theorem delooping hypothesis-theorem stabilization hypothesis-theorem The generalization of the bicategory Span to (∞,n)-categories: An -category of correspondences in ∞-groupoid is an (∞,n)-category whose objects are ∞-groupoids; morphisms are correspondences in ∞Grpd 2-morphisms are correspondences of correspondences (where the triangular sub-diagrams are filled with 2-morph…

In any context it is of interest to ask which kind of morphisms arise as pullbacks along a classifying morphism to some universal object of some universal morphism The Grothendieck construction describes this in the context of Cat: a morphism of categories – i.e. a functor – is called a fibered category or Grothendieck fibration if it is encoded in a pseudofunctor/2-functor . The reconstruction o…

A tensor triangulated category is a category that carries the structure of a symmetric monoidal category and of a triangulated category in a compatible way. There are different variants of the definition in the literature, asking for successively more structure. To start with, a tensor triangulated category must be at least a category equipped with the structure of a symmetric monoidal category (…

For a regular category , the regular coverage on is the coverage in which each covering family has just one element which is a regular epimorphism. The Grothendieck topology generated from a regular coverage is called the regular topology. It is the subcanonical Grothendieck topology whose covering families are generated by single regular epimorphisms: the regular coverage. If is exact or has pul…

There are a number of approaches to apply category theory to probability and related fields, such as statistics, information theory and dynamical systems. On one hand, one can study the existing structures in traditional probability theory (such as probability spaces, integration, and so on) using a categorical lens. For instance, the Giry monad models the formation of spaces of probability measu…

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 the strict sense of the word, a cartesian product is a product in Set, the categor…

Definitions Transfors between 2-categories Morphisms in 2-categories Structures in 2-categories Limits in 2-categories Structures on 2-categories The Kan extension of a functor with respect to a functor is, if it exists, a kind of best approximation to the problem of finding a functor such that hence to extending the domain of through from to . More generally, this notion makes sense not only in …

constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism A thunk-force category is a category that models call-by-value programming languages with effects. More commonly, terms in a call-by-value language are modelled as morphisms in the Kleisli category of a strong monad. A thunk-force category axiomatizes the Kleisli category…