category-theory

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…

A k-tuply monoidal n-category is an $n$-category that has a notion of braiding whose commutativity is parameterised by $k \leq n + 2$ in a suitable sense. E.g. for $n = 2$: a 1-tuply monoidal ...

Alexander Campbell is an Australian higher category theorist. On gerbes in nonabelian cohomology via tricategories: On the straightening theorem: In a model category Quillen equivalent to the canonical model structure on 2-categories:

symmetric monoidal (∞,1)-category of spectra A cartesian monad is a monad on a locally cartesian category that preserves pullbacks and whose unit and multiplication are cartesian natural transformations. Ordinary categories can be defined as monads in the bicategory of spans of sets. Multicategories can be defined in a similar way. (A multicategory is like an ordinary category where each morphism…

A category is cocomplete if it has all small colimits: that is, if every small diagram where is a small category has a colimit in . Equivalently, a category is cocomplete if it has all small wide pushouts and an initial object. The most natural morphisms between cocomplete categories are the cocontinuous functors. A category is cocomplete if and only if it has small coproducts and reflexive coequ…

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 As every topos, a category of presheaves is cartesian closed monoidal. (cartesian clo…

I am currently reading through some Category Theory lecture notes in my spare time and I stumbled across the fact that in the categorical sense, the product metric on a product of metric spaces $(A, ...

homotopy hypothesis-theorem delooping hypothesis-theorem stabilization hypothesis-theorem A -category is a categorification of the notion of an -algebra. An -category is a -category with a -structure that defines an antinatural transformation? from to , where is the conjugate hom-Hilbert space. Introduced in:

homotopy hypothesis-theorem delooping hypothesis-theorem stabilization hypothesis-theorem A concept of -Hilbert spaces is supposed to be a categorification of that of Hilbert spaces, or at least of finite dimensional such: inner product spaces. One way to define this is as a Kapranov-Voevodsky 2-vector space where the hom-functor plays the role of the categorified inner product (Baez 96). In othe…