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
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
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
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
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,
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
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