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Strong monad

In category theory, a strong monad over a monoidal category (C, ⊗, I) is a monad (T, η, μ) together with a natural transformation tA,B : A ⊗ TB → T(A ⊗ B), called (tensorial) strength, such that the d

Tensor-hom adjunction

In mathematics, the tensor-hom adjunction is that the tensor product and hom-functor form an adjoint pair: This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" ref

Monoidal adjunction

Suppose that and are two monoidal categories. A monoidal adjunction between two lax monoidal functors and is an adjunction between the underlying functors, such that the natural transformations and ar

Frobenius reciprocity

In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting. It can be used to leverage knowledge

Free object

In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: th

Reflective subcategory

In mathematics, a full subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector, or localizat

Cotriple homology

In algebra, given a category C with a cotriple, the n-th cotriple homology of an object X in C with coefficients in a functor E is the n-th homotopy group of the E of the augmented simplicial object i

Distributive law between monads

In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other one. Suppose that an

Monad (category theory)

In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor m

Apply

In mathematics and computer science, apply is a function that applies a function to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in

Pseudoalgebra

In algebra, given a 2-monad T in a 2-category, a pseudoalgebra for T is a 2-category-version of algebra for T, that satisfies the laws up to coherent isomorphisms.

Kan extension

Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan,

Adjoint functors

In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two fu

Formal criteria for adjoint functors

In category theory, a branch of mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the following, whic

Quillen adjunction

In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the

Change of rings

In algebra, given a ring homomorphism , there are three ways to change the coefficient ring of a module; namely, for a left R-module M and a left S-module N,
* , the induced module.
* , the coinduce

Beck's monadicity theorem

In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by Jonathan Mock Beck in about 1964. It is often stated in dual

Equivalence of categories

In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are num

Isomorphism of categories

In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D (the identity functor on D) and GF = 1C

Morita equivalence

In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like R, S are Morita equivalent (denoted by ) if

Kleisli category

In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli category is one of two extremal solutions to

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