text:
Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category.

Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

Definition
If $$ F$$ and $$ G$$ are functors between the categories $$ C$$ and $$ D $$ (both from $$ C$$ to $$ D$$), then a natural transformation $$ \eta$$ from $$ F$$ to $$ G$$ is a family of morphisms that satisfies two requirements.
 * 1) The natural transformation must associate, to every object $$ X$$ in $$ C$$, a morphism $$\eta_X :  F(X) \to G(X)$$ between objects of $$ D $$. The morphism $$ \eta_X$$ is called the component of $$ \eta$$ at $$ X $$.
 * 2) Components must be such that for every morphism $$ f :X  \to  Y$$ in $$ C$$ we have:

 * $$\eta_Y \circ F(f) = G(f) \circ \eta_X$$

The last equation can conveniently be expressed by the commutative diagram

If both $$ F$$ and $$ G$$ are contravariant, the vertical arrows in the right diagram are reversed. If $$ \eta$$ is a natural transformation from $$ F$$ to $$ G $$, we also write $$ \eta : F \to G$$ or $$ \eta : F \Rightarrow G $$. This is also expressed by saying the family of morphisms $$ \eta_X: F(X) \to G(X)$$ is natural in $$ X $$.

If, for every object $$ X$$ in $$ C $$, the morphism $$ \eta_X$$ is an isomorphism in $$ D $$, then $$ \eta$$ is said to be a  (or sometimes natural equivalence or isomorphism of functors). Two functors $$ F $$ and $$ G$$ are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from $$ F$$ to $$ G$$.

An infranatural transformation $$ \eta$$ from $$ F$$ to $$ G$$ is simply a family of morphisms $$\eta_X : F(X) \to G(X) $$, for all $$ X$$ in $$ C$$. Thus a natural transformation is an infranatural transformation for which $$ \eta_Y \circ F(f) = G(f) \circ \eta_X$$ for every morphism $$ f : X \to Y $$. The naturalizer of $$ \eta $$, nat$$ (\eta) $$, is the largest subcategory of $$ C$$ containing all the objects of $$ C$$ on which $$ \eta$$ restricts to a natural transformation.

Opposite group
Statements such as
 * "Every group is naturally isomorphic to its opposite group"

abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category $$\textbf{Grp}$$ of all groups with group homomorphisms as morphisms. If $$(G, *)$$ is a group, we define its opposite group $$(G^\text{op}, {*}^\text{op})$$ as follows: $$G^\text{op}$$ is the same set as $$G$$, and the operation $$*^\text{op}$$ is defined by $$a *^\text{op} b = b * a$$. All multiplications in $$G^{\text{op}}$$ are thus "turned around". Forming the opposite group becomes a (covariant) functor from $$\textbf{Grp}$$ to $$\textbf{Grp}$$ if we define $$f^{\text{op}} = f$$ for any group homomorphism $$f: G \to H$$. Note that $$f^\text{op}$$ is indeed a group homomorphism from $$G^\text{op}$$ to $$H^\text{op}$$:
 * $$f^\text{op}(a *^\text{op} b) = f(b * a) = f(b) * f(a) = f^\text{op}(a) *^\text{op} f^\text{op}(b).$$

The content of the above statement is:
 * "The identity functor $$\text{Id}_{\textbf{Grp}}: \textbf{Grp} \to \textbf{Grp}$$ is naturally isomorphic to the opposite functor $${\text{op}}: \textbf{Grp} \to \textbf{Grp}$$"

To prove this, we need to provide isomorphisms $$\eta_G: G \to G^{\text{op}}$$ for every group $$G$$, such that the above diagram commutes. Set $$ \eta_G(a) = a^{-1}$$. The formulas $$(a * b)^{-1} = b^{-1}*a^{-1}= a^{-1}*^{\text{op}} b^{-1}$$ and $$ (a^{-1})^{-1} = a$$ show that $$\eta_G$$ is a group homomorphism with inverse $$ \eta_{G^\text{op}}$$. To prove the naturality, we start with a group homomorphism $$f : G \to H$$ and show $$\eta_H \circ f = f^{\text{op}} \circ \eta_G$$, i.e. $$ (f(a))^{-1} = f^\text{op}(a^{-1})$$ for all $$a$$ in $$G$$. This is true since $$f^{\text{op}} = f$$ and every group homomorphism has the property $$(f(a))^{-1} = f(a^{-1})$$.

Modules
Let $$ \varphi:M \longrightarrow M^{\prime} $$ be an $$R $$-module homomorphism of right modules. For every left module $$ N $$ there is a natural map $$ \varphi \otimes N: M \otimes_{R} N \longrightarrow M^{\prime} \otimes_{R} N$$, form a natural transformation $$\eta: M \otimes_{R} - \implies M' \otimes_{R} - $$. For every right module $$ N $$ there is a natural map $$ \eta_{N}: \text{Hom}_{R}(M',N) \longrightarrow \text{Hom}_{R}(M,N) $$ defined by $$ \eta_{N}(f) = f\varphi$$, form a natural transformation $$ \eta:\text{Hom}_{R}(M',-) \implies \text{Hom}_{R}(M,-) $$.

Abelianization
Given a group $$G$$, we can define its abelianization $$G^{\text{ab}} = G/$$ $[G,G]$. Let $$\pi_G: G \to G^{\text{ab}}$$ denote the projection map onto the cosets of $$[G,G]$$. This homomorphism is "natural in $$G$$", i.e., it defines a natural transformation, which we now check. Let $$H$$ be a group. For any homomorphism $$f : G \to H$$, we have that $$[G,G]$$ is contained in the kernel of $$\pi_H \circ f$$, because any homomorphism into an abelian group kills the commutator subgroup. Then $$\pi_H \circ f$$ factors through $$G^{\text{ab}}$$ as $$f^{\text{ab}} \circ \pi_G = \pi_H \circ f$$ for the unique homomorphism $$f^{\text{ab}} : G^{\text{ab}} \to H^{\text{ab}}$$. This makes $${\text{ab}} : \textbf{Grp} \to \textbf{Grp}$$ a functor and $$\pi$$ a natural transformation, but not a natural isomorphism, from the identity functor to $$\text{ab}$$.

Hurewicz homomorphism
Functors and natural transformations abound in algebraic topology, with the Hurewicz homomorphisms serving as examples. For any pointed topological space $$(X,x)$$ and positive integer $$n$$ there exists a group homomorphism

 * $$h_n \colon \pi_n(X,x) \to H_n(X)$$

from the $$n$$-th homotopy group of $$(X,x)$$ to the $$n$$-th homology group of $$X$$. Both $$\pi_n$$ and $$H_n$$ are functors from the category Top* of pointed topological spaces to the category Grp of groups, and $$h_n$$ is a natural transformation from $$\pi_n$$ to $$H_n$$.

Determinant
Given commutative rings $$R$$ and $$S$$ with a ring homomorphism $$f : R \to S$$, the respective groups of invertible $$n \times n$$ matrices $$\text{GL}_n(R)$$ and $$\text{GL}_n(S)$$ inherit a homomorphism which we denote by $$\text{GL}_n(f)$$, obtained by applying $$f$$ to each matrix entry. Similarly, $$f$$ restricts to a group homomorphism $$f^* : R^* \to S^*$$, where $$R^*$$ denotes the group of units of $$R$$. In fact, $$\text{GL}_n$$ and $$*$$ are functors from the category of commutative rings $$\textbf{CRing}$$ to $$\textbf{Grp}$$. The determinant on the group $$\text{GL}_n(R)$$, denoted by $$\text{det}_R$$, is a group homomorphism

 * $$\mbox{det}_R \colon \mbox{GL}_n(R) \to R^*$$

which is natural in $$R$$: because the determinant is defined by the same formula for every ring, $$f^* \circ \text{det}_R = \text{det}_S\circ \text{GL}_n(f)$$ holds. This makes the determinant a natural transformation from $$\text{GL}_n$$ to $$*$$.

Double dual of a vector space
For example, if $$K$$ is a field, then for every vector space $$V$$ over $$K$$ we have a "natural" injective linear map $$V \to V^{**}$$ from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps are the components of a natural transformation from the identity functor to the double dual functor.

Finite calculus
For every abelian group $$G$$, the set $$\text{Hom}_\textbf{Set}(\mathbb{Z}, U(G))$$ of functions from the integers to the underlying set of $$G$$ forms an abelian group $$V_{\mathbb{Z}}(G)$$ under pointwise addition. (Here $$U$$ is the standard forgetful functor $$U:\textbf{Ab} \to \textbf{Set}$$.) Given an $$\textbf{Ab}$$ morphism $$\varphi: G \to G' $$, the map $$V_\mathbb{Z}(\varphi): V_\mathbb{Z}(G) \to V_\mathbb{Z}(G')$$ given by left composing $$\varphi$$ with the elements of the former is itself a homomorphism of abelian groups; in this way we obtain a functor $$V_{\mathbb{Z}}: \textbf{Ab} \to \textbf{Ab}$$. The finite difference operator $$\Delta_G$$ taking each function $$f: \mathbb{Z} \to U(G)$$ to $$\Delta(f): n \mapsto f(n+ 1) - f(n)$$ is a map from $$V_{\mathbb{Z}}(G)$$ to itself, and the collection $$\Delta$$ of such maps gives a natural transformation $$\Delta: V_\mathbb{Z} \to V_\mathbb{Z}$$.

Tensor-hom adjunction
Consider the category $\textbf{Ab}$ of abelian groups and group homomorphisms. For all abelian groups $$X$$, $$Y$$ and $$Z$$ we have a group isomorphism
 * $$\text{Hom}(X \otimes Y, Z) \to \text{Hom}(X, \text{Hom}(Y, Z))$$.

These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors $$\textbf{Ab}^{\text{op}} \times \textbf{Ab}^{\text{op}} \times \textbf{Ab} \to \textbf{Ab}$$. (Here "op" is the opposite category of $$\textbf{Ab}$$, not to be confused with the trivial opposite group functor on $$\textbf{Ab}$$ !)

This is formally the tensor-hom adjunction, and is an archetypal example of a pair of adjoint functors. Natural transformations arise frequently in conjunction with adjoint functors, and indeed, adjoint functors are defined by a certain natural isomorphism. Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit.

Unnatural isomorphism
The notion of a natural transformation is categorical, and states (informally) that a particular map between functors can be done consistently over an entire category. Informally, a particular map (esp. an isomorphism) between individual objects (not entire categories) is referred to as a "natural isomorphism", meaning implicitly that it is actually defined on the entire category, and defines a natural transformation of functors; formalizing this intuition was a motivating factor in the development of category theory.

Conversely, a particular map between particular objects may be called an unnatural isomorphism (or "an isomorphism that is not natural") if the map cannot be extended to a natural transformation on the entire category. Given an object $$X,$$ a functor $$G$$ (taking for simplicity the first functor to be the identity) and an isomorphism $$\eta\colon X \to G(X),$$ proof of unnaturality is most easily shown by giving an automorphism $$A\colon X \to X$$ that does not commute with this isomorphism (so $$\eta \circ A \neq G(A) \circ \eta$$). More strongly, if one wishes to prove that $$X$$ and $$G(X)$$ are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for any isomorphism $$\eta$$, there is some $$A$$ with which it does not commute; in some cases a single automorphism $$A$$ works for all candidate isomorphisms $$\eta$$ while in other cases one must show how to construct a different $$A_\eta$$ for each isomorphism. The maps of the category play a crucial role – any infranatural transform is natural if the only maps are the identity map, for instance.

This is similar (but more categorical) to concepts in group theory or module theory, where a given decomposition of an object into a direct sum is "not natural", or rather "not unique", as automorphisms exist that do not preserve the direct sum decomposition – see for example.

Some authors distinguish notationally, using $$\cong$$ for a natural isomorphism and $$\approx$$ for an unnatural isomorphism, reserving $$= $$ for equality (usually equality of maps).

Example: fundamental group of torus
As an example of the distinction between the functorial statement and individual objects, consider homotopy groups of a product space, specifically the fundamental group of the torus.

The homotopy groups of a product space are naturally the product of the homotopy groups of the components, $$\pi_n((X,x_0) \times (Y,y_0)) \cong \pi_n((X,x_0)) \times \pi_n((Y,y_0)),$$ with the isomorphism given by projection onto the two factors, fundamentally because maps into a product space are exactly products of maps into the components – this is a functorial statement.

However, the torus (which is abstractly a product of two circles) has fundamental group isomorphic to $$Z^2$$, but the splitting $$\pi_1(T,t_0) \approx \mathbf{Z} \times \mathbf{Z}$$ is not natural. Note the use of $$\approx$$, $$\cong$$, and $$=$$:
 * $$\pi_1(T,t_0) \approx \pi_1(S^1,x_0) \times \pi_1(S^1,y_0) \cong \mathbf{Z} \times \mathbf{Z} = \mathbf{Z}^2.$$

This abstract isomorphism with a product is not natural, as some isomorphisms of $$T$$ do not preserve the product: the self-homeomorphism of $$T$$ (thought of as the quotient space $$R^2/\mathbb{Z}^2$$) given by $$\left(\begin{smallmatrix}1 & 1\\0 & 1\end{smallmatrix}\right)$$ (geometrically a Dehn twist about one of the generating curves) acts as this matrix on $$\mathbb{Z}^2$$ (it's in the general linear group $$\text{GL}(\mathbb{Z}, 2)$$ of invertible integer matrices), which does not preserve the decomposition as a product because it is not diagonal. However, if one is given the torus as a product $$(T,t_0) = (S^1,x_0) \times (S^1,y_0)$$ – equivalently, given a decomposition of the space – then the splitting of the group follows from the general statement earlier. In categorical terms, the relevant category (preserving the structure of a product space) is "maps of product spaces, namely a pair of maps between the respective components".

Naturality is a categorical notion, and requires being very precise about exactly what data is given – the torus as a space that happens to be a product (in the category of spaces and continuous maps) is different from the torus presented as a product (in the category of products of two spaces and continuous maps between the respective components).

Example: dual of a finite-dimensional vector space
Every finite-dimensional vector space is isomorphic to its dual space, but there may be many different isomorphisms between the two spaces. There is in general no natural isomorphism between a finite-dimensional vector space and its dual space. However, related categories (with additional structure and restrictions on the maps) do have a natural isomorphism, as described below.

The dual space of a finite-dimensional vector space is again a finite-dimensional vector space of the same dimension, and these are thus isomorphic, since dimension is the only invariant of finite-dimensional vector spaces over a given field. However, in the absence of additional constraints (such as a requirement that maps preserve the chosen basis), the map from a space to its dual is not unique, and thus such an isomorphism requires a choice, and is "not natural". On the category of finite-dimensional vector spaces and linear maps, one can define an infranatural isomorphism from vector spaces to their dual by choosing an isomorphism for each space (say, by choosing a basis for every vector space and taking the corresponding isomorphism), but this will not define a natural transformation. Intuitively this is because it required a choice, rigorously because any such choice of isomorphisms will not commute with, say, the zero map; see for detailed discussion.

Starting from finite-dimensional vector spaces (as objects) and the identity and dual functors, one can define a natural isomorphism, but this requires first adding additional structure, then restricting the maps from "all linear maps" to "linear maps that respect this structure". Explicitly, for each vector space, require that it comes with the data of an isomorphism to its dual, $$\eta_V\colon V \to V^*$$. In other words, take as objects vector spaces with a nondegenerate bilinear form $$b_V\colon V \times V \to K$$. This defines an infranatural isomorphism (isomorphism for each object). One then restricts the maps to only those maps $$T\colon V \to U$$ that commute with the isomorphisms: $$T^*(\eta_{U}(T(v))) = \eta_{V}(v)$$ or in other words, preserve the bilinear form: $$b_{U}(T(v),T(w))=b_V(v,w)$$. (These maps define the naturalizer of the isomorphisms.) The resulting category, with objects finite-dimensional vector spaces with a nondegenerate bilinear form, and maps linear transforms that respect the bilinear form, by construction has a natural isomorphism from the identity to the dual (each space has an isomorphism to its dual, and the maps in the category are required to commute). Viewed in this light, this construction (add transforms for each object, restrict maps to commute with these) is completely general, and does not depend on any particular properties of vector spaces.

In this category (finite-dimensional vector spaces with a nondegenerate bilinear form, maps linear transforms that respect the bilinear form), the dual of a map between vector spaces can be identified as a transpose. Often for reasons of geometric interest this is specialized to a subcategory, by requiring that the nondegenerate bilinear forms have additional properties, such as being symmetric (orthogonal matrices), symmetric and positive definite (inner product space), symmetric sesquilinear (Hermitian spaces), skew-symmetric and totally isotropic (symplectic vector space), etc. – in all these categories a vector space is naturally identified with its dual, by the nondegenerate bilinear form.

Vertical composition
If $$\eta : F \Rightarrow G$$ and $$\epsilon: G \Rightarrow H$$ are natural transformations between functors $$F, G, H: C \to D$$, then we can compose them to get a natural transformation $$\epsilon \circ \eta: F \Rightarrow H$$. This is done componentwise:
 * $$(\epsilon \circ \eta)_X = \epsilon_X \circ \eta_X$$.

This vertical composition of natural transformations is associative and has an identity, and allows one to consider the collection of all functors $$C \to D$$ itself as a category (see below under Functor categories). The identity natural transformation $$\mathrm{id}_F$$ on functor $$F$$ has components $$(\mathrm{id}_F)_X = \mathrm{id}_{F(X)}$$.
 * For $$\eta : F \Rightarrow G$$, $$\mathrm{id}_G \circ \eta = \eta = \eta \circ \mathrm{id}_F$$.

Horizontal composition
If $$\eta: F \Rightarrow G$$ is a natural transformation between functors $$F, G: C \to D$$ and $$\epsilon: J \Rightarrow K$$ is a natural transformation between functors $$J, K: D \to E$$, then the composition of functors allows a composition of natural transformations $$\epsilon * \eta: J \circ F \Rightarrow K \circ G$$ with components
 * $$(\epsilon * \eta)_X = \epsilon_{G(X)} \circ J(\eta_X) = K(\eta_X) \circ \epsilon_{F(X)}$$.

By using whiskering (see below), we can write
 * $$(\epsilon * \eta)_X = (\epsilon G)_X \circ (J \eta)_X = (K \eta)_X \circ (\epsilon F)_X$$,

hence
 * $$\epsilon * \eta = \epsilon G \circ J \eta = K \eta \circ \epsilon F$$.

This horizontal composition of natural transformations is also associative with identity. This identity is the identity natural transformation on the identity functor, i.e., the natural transformation that associate to each object its identity morphism: for object $$X$$ in category $$C$$, $$(\mathrm{id}_{\mathrm{id}_C})_X = \mathrm{id}_{\mathrm{id}_C(X)} = \mathrm{id}_X$$.
 * For $$\eta: F \Rightarrow G$$ with $$F, G: C \to D$$, $$\mathrm{id}_{\mathrm{id}_D} * \eta = \eta = \eta * \mathrm{id}_{\mathrm{id}_C}$$.

As identity functors $$\mathrm{id}_C$$ and $$\mathrm{id}_D$$ are functors, the identity for horizontal composition is also the identity for vertical composition, but not vice versa.

Whiskering
Whiskering is an external binary operation between a functor and a natural transformation.

If $$\eta: F \Rightarrow G$$ is a natural transformation between functors $$F, G: C \to D$$, and $$H: D \to E$$ is another functor, then we can form the natural transformation $$H \eta: H \circ F \Rightarrow H \circ G$$ by defining
 * $$(H \eta)_X = H(\eta_X)$$.

If on the other hand $$K: B \to C$$ is a functor, the natural transformation $$\eta K: F \circ K \Rightarrow G \circ K$$ is defined by
 * $$(\eta K)_X = \eta_{K(X)}$$.

It's also an horizontal composition where one of the natural transformations is the identity natural transformation:
 * $$H \eta = \mathrm{id}_H * \eta$$ and $$\eta K = \eta * \mathrm{id}_K$$.

Note that $$\mathrm{id}_H$$ (resp. $$\mathrm{id}_K$$) is generally not the left (resp. right) identity of horizontal composition $$*$$ ($$H \eta \neq \eta$$ and $$\eta K \neq \eta$$ in general), except if $$H$$ (resp. $$K$$) is the identity functor of the category $$D$$ (resp. $$C$$).

Interchange law
The two operations are related by an identity which exchanges vertical composition with horizontal composition: if we have four natural transformations $$\alpha, \alpha', \beta, \beta'$$ as shown on the image to the right, then the following identity holds:
 * $$ (\beta' \circ \alpha') * (\beta \circ \alpha) = (\beta' * \beta) \circ (\alpha' * \alpha)$$.

Vertical and horizontal compositions are also linked through identity natural transformations:
 * for $$F: C \to D$$ and $$G: D \to E$$, $$\mathrm{id}_G * \mathrm{id}_F = \mathrm{id}_{G \circ F}$$.

As whiskering is horizontal composition with an identity, the interchange law gives immediately the compact formulas of horizontal composition of $$\eta: F \Rightarrow G$$ and $$\epsilon: J \Rightarrow K$$ without having to analyze components and the commutative diagram:
 * $$\begin{align}

\epsilon * \eta & = (\epsilon \circ \mathrm{id}_J) * (\mathrm{id}_G \circ \eta) = (\epsilon * \mathrm{id}_G) \circ (\mathrm{id}_J * \eta) = \epsilon G \circ J \eta \\ & = (\mathrm{id}_K \circ \epsilon) * (\eta \circ \mathrm{id}_F) = (\mathrm{id}_K * \eta) \circ (\epsilon * \mathrm{id}_F) = K \eta \circ \epsilon F \end{align}$$.

Functor categories
If $$C$$ is any category and $$I$$ is a small category, we can form the functor category $$C^I$$ having as objects all functors from $$I$$ to $$C$$ and as morphisms the natural transformations between those functors. This forms a category since for any functor $$F$$ there is an identity natural transformation $$1_F: F \to F$$ (which assigns to every object $$X$$ the identity morphism on $$F(X)$$) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation.

The isomorphisms in $$C^I$$ are precisely the natural isomorphisms. That is, a natural transformation $$\eta: F \to G$$ is a natural isomorphism if and only if there exists a natural transformation $$\epsilon: G \to F$$ such that $$\eta\epsilon = 1_G$$ and $$\epsilon\eta = 1_F$$.

The functor category $$C^I$$ is especially useful if $$I$$ arises from a directed graph. For instance, if $$I$$ is the category of the directed graph, then $$C^I$$ has as objects the morphisms of $$C$$, and a morphism between $$\phi: U \to V$$ and $$\psi: X \to Y$$ in $$C^I$$ is a pair of morphisms $$f: U \to X$$ and $$g: V \to Y$$ in $$C$$ such that the "square commutes", i.e. $$\psi \circ f = g \circ \phi$$.

More generally, one can build the 2-category $$\textbf{Cat}$$ whose The horizontal and vertical compositions are the compositions between natural transformations described previously. A functor category $$C^I$$ is then simply a hom-category in this category (smallness issues aside).
 * 0-cells (objects) are the small categories,
 * 1-cells (arrows) between two objects $$C$$ and $$D$$ are the functors from $$C$$ to $$D$$,
 * 2-cells between two 1-cells (functors) $$F:C\to D$$ and $$G:C\to D$$ are the natural transformations from $$F$$ to $$G$$.

More examples
Every limit and colimit provides an example for a simple natural transformation, as a cone amounts to a natural transformation with the diagonal functor as domain. Indeed, if limits and colimits are defined directly in terms of their universal property, they are universal morphisms in a functor category.

Yoneda lemma
If $$X$$ is an object of a locally small category $$C$$, then the assignment $$Y \mapsto \text{Hom}_{C}(X, Y)$$ defines a covariant functor $$F_X: C \to \textbf{Set}$$. This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of $$X$$). The natural transformations from a representable functor to an arbitrary functor $$F: C \to \textbf{Set}$$ are completely known and easy to describe; this is the content of the Yoneda lemma.

Historical notes
Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations.

The context of Mac Lane's remark was the axiomatic theory of homology. Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly would be isomorphic to those of the singular theory. What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups.

title: Natural transformation


text:
User:Toby Bartels

I am Toby Bartels (they/them).

For a few months when I first arrived, I edited under the username, which is now archived as. I am not the current user of the name, and I have nothing to do with the previously proposed optional censorship protocol called that.

No rights reserved
I reserve no copyright or patent rights to any of my creative work. As far as copyright and patent law are concerned, I grant you permission to do anything with it that you wish. See  for more information.

This is a perfectly free, noncopyleft licence. Because it is all-permissive, it is automatically compatible with any other licence. If you're keeping track of Wikipedians whose works are compatible with other licences, then feel free to add templates and category headings to this effect below.

It is not possible for me to place in the public domain anything that I do not have complete rights to. For the most part, therefore, my work on Wikipedia cannot be in the public domain. Given the permission above, you shouldn't need anything to be in the public domain anyway. However, if you want me to place something explicitly in the public domain, then ask me, and I will do so if it is possible.

Viewing deleted pages
It has always been a bad idea to hide most deleted pages from ordinary readers who want to see them, and too much gets deleted in the first place (although this has varied over time). I won't violate established policy, but I'll do anything short of that to provide you with any information that you may want about deleted revisions of any page for any reason. (I only have the power to do this on the English Wikipedia.)

Note that I am not very active compared to some other Administrators, and you can usually get what you need at Requests for undeletion. However, if that is not working for you, then feel free to contact me on my talk page or at  (an address that may change if it attracts more spam).

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Toby Bartels


text:
Mongol invasion of Kievan Rus'

The Mongol Empire invaded and conquered much of Kievan Rus' in the mid-13th century, sacking numerous cities including the largest such as Kiev (50,000 inhabitants) and Chernigov (30,000 inhabitants). The siege of Kiev in 1240 by the Mongols is generally held to mark the end of Kievan Rus', which had been undergoing fragmentation. Many other principalities and urban centres in the northwest and southwest escaped complete destruction or suffered little to no damage from the Mongol invasion, including Galicia–Volhynia, Novgorod, Pskov, Smolensk, Polotsk, Vitebsk, and probably Rostov and Uglich.

The campaign was heralded by the Battle of the Kalka River in May 1223, which resulted in a Mongol victory over the forces of several principalities as well as the remnants of the Cumans under Köten. The Mongols retreated, having gathered their intelligence, which was the purpose of the reconnaissance-in-force. A full-scale invasion by Batu Khan followed, with most of Kievan Rus' overran in 1237–1238. The Mongols captured Kiev in 1240 and moved west into Hungary and Poland.

The invasion was ended by the Mongol succession process upon the death of Ögedei Khan. Even those principalities who avoided physical conquest, were eventually forced to accept Mongol supremacy in the form of tribute – as in the case of Galicia-Volhynia, Polotsk and Novgorod – if not outright vassalage, of the Golden Horde, until well into the 14th century. Although a Russian army defeated the Mongols at the Battle of Kulikovo in 1380, the demands of tribute from Russian princes continued until about 1480.

Background
The Mongols had plans to conquer Eastern Europe long before Batu's campaign in 1237. In 1207, Genghis Khan sent his eldest son Jochi to conquer the tribes north of the Selenga River and in the Irtysh valley, which included the lands of Eastern Europe in the Jochi Ulus. However, these plans were not implemented during the lifetime of Genghis Khan. In 1222–1224, Subedei and Jebe conducted a campaign with a 30,000-strong army in Transcaucasia and Southeastern Europe, which is traditionally considered reconnaissance. However, the Secret History and Rashid ad-Din state that the purpose of this campaign was to be supported by the forces of Jochi and included Cumans, Alans, Hungary, and Rus', including Kiev. The kurultai of 1235, after which the invasion of Europe took place, repeated these goals. The campaign of Subedei and his 30,000 troops to the Caspian steppes was a new step in the conquest of Eastern Europe, and it began in the late 1220s. In 1235, a kurultai was convened to outline a general Mongol campaign, in which the troops of other uluses were to take part. Ogedei sent Batu, Buri, Munk, and other princes on a campaign to help Subetai, and each Chingizid led with him one or more tumens of the army. The Mongols prepared for an offensive in 1235 and early 1236 and subjugated the Bashkir tribes, who were forced to allocate several detachments to the Mongol army. The Mongols concentrated in the Caspian steppes in the autumn of 1236 under the general leadership of Jochi's son Batu. The first blow of the united Chingizid army hit the Volga Bulgaria.

The Mongols (called "Tartars" in contemporary accounts) had already defeated a united Rus' army led by Mstislav the Bold and Mstislav Romanovich the Old on 1 April 1223, at the Battle of the Kalka River. In 1237, they besieged and took Ryazan, on Vladimir fell in early February 1238. "For our sins", wrote a chronicler, "unknown nations arrived. No one knew their origin or whence they came, or what religion they practiced. That is known only to God, and perhaps to wise men learned in books". Although this defeat left the principalities at the mercy of invaders, the Mongol or Tartar forces retreated and did not reappear for thirteen years, during which time the princes of Rus' went on quarreling and fighting as before, until they were startled by a new and much more formidable invading force. In The Secret History of the Mongols, the only reference to this early battle is:

"Then he (Chinggis Khan) sent Dorbei the Fierce off against the city of Merv, and on to conquer the people between Iraq and the Indus. He sent Subetei the Brave off to war in the North where he defeated eleven kingdoms and tribes, crossing the Volga and Ural Rivers, finally going to war with Kiev."

The Secret History of the Mongols reports that Ogedei sent Batu, Buri, Munk, and many other princes on a campaign to help Subetai, who was facing a strong resistance from various peoples and cities under Genghis Khan's command. The list of Genghisides who participated in the campaign is present in works such as The Secret Legend, Yuan Shi, and The Collection of Chronicles by the Persian historian Rashid ad-Din. In addition to Batu, other Chingizids who participated in the campaign included the sons of Jochi, Orda-Ezhen, Shiban, Tangkut, and Berke; the grandson of Chagatai, Buri, and the son of Chagatai, Baydar; the sons of Ogedei, Guyuk, and Kadan; the sons of Tolui, Munke, and Buchek; the son of Genghis Khan, Külkhan, and the grandson of Genghis Khan's brother, Argasun. In 1235 and early 1236, the assembled army prepared for an offensive, and then subjugated the Bashkir tribes, who were forced to allocate several detachments to the Mongol army. In the autumn of 1236, the Mongols concentrated in the Caspian steppes under the general leadership of Jochi's son Batu.

The first blow of the united Chingizid army hit the Volga Bulgaria. Until the mid-1220s, the Volga Bulgaria was in constant conflict with the Vladimir-Suzdal and Muromo-Ryazan principalities. The parties undertook campaigns, there were constant skirmishes, and the victories in which were mainly won by Rus' troops. However, with the appearance of the Mongols at their borders, the Bulgars began to seek peace, which was met with understanding and support from the Rus' princes. Over the course of several years, the Rus' and the Bulgars normalized relations, which allowed the Volga Bulgaria to devote all its forces to preparing to repel the alleged Mongol invasion. Ramparts were created in the forests that covered the main cities, the cities themselves were fortified, and the garrisons increased. However, all these measures were in vain – the Volga Bulgaria was defeated with lightning speed and completely conquered by the spring of 1237.

The next stage of the campaign was an attack on the Polovtsians and Alans. From the Lower Volga region, the Mongols moved on a broad front to the mouth of the Don, where another concentration of troops took place. The offensive continued until the autumn of 1237 and ended with the defeat of the Polovtsians and Alans. After that, the Mongols captured the lands of the Burtases, Moksha, and Erzi. The grandiose Zolotarevskoe battle took place near a strategic crossing over the Sura River. According to the historian Kargalov, the fighting in 1237 was undertaken to create a springboard for a campaign against Rus'. By the end of the year, a huge Mongol army and detachments allied with Batu stood on the borders of Russia. Preparations for a winter campaign against North-Eastern Rus' by the Mongols began in the autumn of 1237. Their troops were grouped near Voronezh, and detachments that had previously fought with the Polovtsy and Alans were drawn there.

Invasion of Batu Khan

The vast Mongolian Great Khanate army of around 120,000 mounted archers, commanded by Batu Khan and Subutai, crossed the Volga River and invaded Volga Bulgaria in late 1236. It took them only a month to extinguish the resistance of the Volga Bulgars, the Cumans-Kipchaks and the Alans.

Immediately prior to the invasion, Friar Julian from Hungary had travelled to the eastern border of the Rus' and learned of the Mongol army, which was waiting for the onset of winter so that they could cross the frozen rivers and swamps. In his letter to the Pope's legate in Hungary, Julian described meeting Mongol messengers who had been detained by Yuri II of Vladimir on their way to Hungary. Yuri II gave their letter to Julian.

In November 1237, Batu Khan sent his envoys to the court of Yuri II and demanded his submission. According to the Laurentian Codex, the Mongols actually came seeking peace, but Yuri II treated them with disdain:

"As they did before, the messengers came, those evil bloodsuckers, saying: 'Make peace with us'. He did not want that, as the prophet said: 'Glorious war is better than disgraceful peace'. These godless men with their deceitful peace will cause great dismay to our lands, as they have already done much evil here."

Regardless of what impression Yuri II may have given the Mongol delegations, of which several are mentioned, he did his best to avoid direct conflict. He sent them away with what were described as gifts, which were essentially tribute or bribes to keep them from invading.

The Mongols attacked from several directions. One section attacked Suzdal, one from the Volga, and another from the south towards Ryazan. According to Rashid al-Din Hamadani, the Siege of Ryazan was conducted by Batu, Orda, Güyük, Mengu Qa'an, Kulkan, Kadan, and Buri. The city fell after three days. Alarmed by the news, Yuri II sent his sons to detain the invaders, but they were defeated and ran for their lives. Yuri II also fled Vladimir for Yaroslavl.

Having burnt down Kolomna and Moscow, the horde laid siege to Vladimir on 4 February 1238. Three days later, the capital of Vladimir-Suzdal was taken and burnt to the ground. The royal family perished in the fire, while the grand prince retreated northward. Crossing the Volga, Vladimir mustered a new army, which was encircled and totally annihilated by the Mongols in the Battle of the Sit River on 4 March.

"And the Tartars took the town [of Ryazan] on December 21... They likewise killed the [Prince] and Knyaginya, and men, women and children, monks, nuns and priests, some by fire, some by the sword and violated nuns, priests' wives, good women and girls in the presence of their mothers and sisters."

Thereupon Batu Khan divided his army into smaller units, which ransacked fourteen cities of northeastern Rus': Rostov, Uglich, Yaroslavl, Kostroma, Kashin, Ksnyatin, Gorodets, Galich, Pereslavl-Zalessky, Yuriev-Polsky, Dmitrov, Volokolamsk, Tver, and Torzhok. Chinese siege engines were used by the Mongols under Tului to raze the walls of many cities. The most difficult to take was the small town of Kozelsk, whose boy-prince Vasily, son of Titus, and inhabitants resisted the Mongols for seven weeks, killing 4,000. As the story goes, at the news of the Mongol approach, the whole town of Kitezh with all its inhabitants was submerged into a lake, where, as legend has it, it may be seen to this day. Major principalities and urban centres which escaped destruction or suffered little to no damage from the Mongol invasion included Novgorod, Pskov, Smolensk, Polotsk, Vitebsk, and probably Rostov and Uglich. The Mongols planned to advance on Novgorod, but the principality was spared the fate of its brethren by the decision to preemptively surrender. In mid-1238, Batu Khan devastated the Crimea and pacified Mordovia. In the winter of 1239, he sacked Chernigov and Pereyaslavl.

While Kiev and its grand prince was still formally acknowledged as senior amongst the principalities of Rus', frequent internecine dynastic feuding between rival claimants had left the city weakened. Indeed, by the time Kiev fell to the Mongols, the head of the city's defenses owed allegiance to Prince Daniel of Galicia-Volhynia. Prince Daniel had taken Kiev under his protection the previous year by arrangement with Prince Michael of Kiev, who fled after originally resisting the Mongols, then losing to them his main stronghold, Chernigov.

The Mongols approach on Kiev in November 1240 apparently made a grim impression upon its defenders. The chronicler writes: “And nothing could be heard above the squeaking of his carts, the bawling of his [Batu’s] innumerable camels, and the neighing of his herds of horses, and the Land of Rus’ was full of enemies." After many days of siege, the horde stormed Kiev in December 1240. The city was ransacked and pillaged immensely, although the building of St. Sophia Cathedral survived intact. Historian Serhii Plokhy relates the description of one Giovanni da Pian del Carpine, an ambassador of Pope Innocent IV who passed through Kiev in six years later: "When we were journeying through that land, we came across countless skulls and bones of dead men lying about on the ground".

Batu Khan's forces went on to ravage much of Galicia-Volhynia, despite Daniel's fierce resistance. The Tartars then resolved to "reach the ultimate sea", where they could proceed no further and invaded Hungary (under Batu Khan) and Poland (under Baidar and Kaidu). Batu Khan captured Pest, and then on Christmas Day 1241, Esztergom.

The principalities became part of the Jochid appanage ruled by Batu. Batu sited a semi-nomadic capital, called Sarai or Sarai Batu (Batu's Palaces), on the lower Volga. The Jochid appanage came to be known as the Golden Horde.

Impact on development
Giovanni de Plano Carpini, the pope's envoy to the Mongol great khan, traveled through Kiev in February 1246 and wrote:

"They [the Mongols] attacked Russia, where they made great havoc, destroying cities and fortresses and slaughtering men; and they laid siege to Kiev, the capital of Russia; after they had besieged the city for a long time, they took it and put the inhabitants to death. When we were journeying through that land we came across countless skulls and bones of dead men lying about on the ground. Kiev had been a very large and thickly populated town, but now it has been reduced almost to nothing, for there are at the present time scarce two hundred houses there and the inhabitants are kept in complete slavery."

The influence of the Mongol invasion on the territories was uneven. Colin McEvedy estimates the population dropped from 7.5 million prior to the invasion to seven million afterwards. Centres such as Kiev took centuries to rebuild and recover from the devastation of the initial attack. The Novgorod Republic continued to prosper, and a new entity, Moscow, began to flourish under the Mongols.

Moscow's eventual dominance in northeastern Rus' was in large part attributable to the Mongols. Moscow drew people and wealth, developed trade links, and established a autocratic political system which exerted a powerful influence on Russian society. After the prince of Tver led an uprising in 1327, the rival prince Ivan I of Moscow joined the Mongols in crushing Tver and devastating its lands. By doing so, he eliminated his rival, allowed the Russian Orthodox Church to move its headquarters to Moscow, and was granted the title of grand prince by the Mongols.

As such, the Muscovite prince became the chief intermediary between the Mongol overlords and the Russian princes, which paid further dividends for Moscow's rulers. In the 14th century, the Muscovite princes began "gathering Russian lands" to increase its population and wealth. While the Mongols often raided other territories, they tended to respect the lands controlled by their principal collaborator. This, in turn, attracted nobles and their servants who sought to settle in the relatively secure and peaceful lands of Moscow. Although a Russian army defeated the Golden Horde at the Battle of Kulikovo in 1380, the Mongol domination of Russian-inhabited territories, with the requisite demands of tribute, continued until the Great Stand on the Ugra River in 1480.

Decline of cities
The invasion had significant consequences on the population. Many cities and fortified points were wiped out, with only a fraction surviving. The once flourishing cities of Kiev, Novgorod, and Vladimir suffered a sharp decline, with their populations shrinking to a fraction of what they were before the invasion. For example, Kiev, which had around 50,000 people, had only about 200 houses left after the invasion, according to the papal legate Plano Carpini.

The decline of cities was also accompanied by a decline in culture, crafts, and trade. The pre-Mongol period was considered the heyday of culture, crafts, and trade in ancient Rus', but after the invasion, many cities fell into decay, and stone construction was halted for a long time. Economic ties between cities and surrounding villages were severed, and it took more than 100 years for Russian cities to recover from the invasion of Batu Khan and its consequences.

The destruction of cities and the decline in culture and economy had long-term consequences for Russia. The country was left behind in terms of economic development, and it took centuries for it to catch up with the rest of Europe. The Mongol-Tatar invasion also had a significant impact on Russia's political development, as it paved the way for the emergence of the centralized Moscow state, which gradually absorbed other principalities and became the dominant power in Russia. Overall, the invasion of Batu Khan had a profound and lasting impact on the history of Russia.

Economic setbacks
Stone construction in Russian cities practically ceased for several decades. The production of complex crafts, such as glass jewelry, cloisonne enamel, niello, granulation, and polychrome glazed ceramics stopped. As a result, the Russian handicraft industry regressed several centuries, while the guild industry in the West progressed to the era of primitive accumulation. The Russian handicraft industry had to reacquire the gains that had been made before the invasion.

Population migration
As a result of the invasion many people were forced to flee in front of the advancing tumens of Batu, and in northeastern Rus', residents of the Vladimir-Suzdal and Ryazan principalities sought refuge in more northern lands beyond the Volga. Others fled to sparsely populated areas, taking refuge in dense forests. However, after the departure of the Mongol-Tatars, most of them returned to their former places of residence.

In fact, just a year after the fall of the Vladimir-Suzdal principality, the number of returnees was so great that Prince Yaroslav Yaroslavich was able to gather a large army among them for a campaign against the Lithuanians. Meanwhile, the inhabitants of South Russia's principalities, such as Kiev, Pereyaslav, and Chernigov, fled to northeastern Russia immediately after the invasion.

However, this was not the end of population migrations. Vladimir, Suzdal, Pereslavl-Zalessky, and other cities of northeastern Rus' were repeatedly targeted by Mongol-Tatar campaigns in the second half of the 13th century. As a result, many of their inhabitants gradually moved either to the vicinity of Moscow and Tver, or to the north in regions such as Yaroslavl, Galich, Veliky Ustyug, and more.

Influence on society
Historians have debated the long-term influence of Mongol rule on society. The Mongols have been blamed for the destruction of Kievan Rus' and the breakup of a "Russian" nationality into three components, and the introduction of the concept of "oriental despotism" into Russia. However, most historians agree that Kievan Rus' was not a homogenous entity in a political, cultural, or ethnic sense, and that the invasions only accelerated the period of feudal fragmentation that had begun prior to the invasions. Historians also credit the Mongol regime with an important role in the development of Muscovy as a state. Under Mongol occupation, for example, Muscovy developed its mestnichestvo hierarchy, postal road network (based on Mongolian ortoo system, known in Russian as "yam", hence the terms yamshchik, Yamskoy Prikaz, etc.), census, fiscal system and military organization.

The period of Mongol rule over the former Rus' polities included significant cultural and interpersonal contacts between the Slavic and Mongolian ruling classes. By 1450, the Tatar language had become fashionable in the court of the Grand Prince of Moscow, Vasily II, who was accused of excessive love of the Tatars and their speech, and many Russian noblemen adopted Tatar surnames (for example, a member of the Veliamanov family adopted the Turkic name "Aksak" and his descendants were the Aksakovs).

Many Russian boyar (noble) families traced their descent from the Mongols or Tatars, including Veliaminov-Zernov, Godunov, Arseniev, Bakhmetev, Bulgakov (descendants of Bulgak) and Chaadaev (descendants of Genghis Khan's son Chagatai Khan). In a survey of Russian noble families of the 17th century, over 15% of the Russian noble families had Tatar or Oriental origins.

The Mongols brought about changes in the economic power of states and overall trade. In the religious sphere, St. Paphnutius of Borovsk was the grandson of a Mongol baskak, or tax collector, while a nephew of Khan Bergai of the Golden Horde converted to Christianity and became known as the monk St. Peter Tsarevich of the Horde.

In the judicial sphere, under Mongol influence capital punishment, which during the times of Kievan Rus' had only been applied to slaves, became widespread, and the use of torture became a regular part of criminal procedure. Specific punishments introduced in Moscow included beheading for alleged traitors and branding of thieves (with execution for a third arrest).

Historiography
According to Charles J. Halperin (2011), Fomenko and Nosovskii's popular pseudohistorical Novaia khronologiia (New Chronology), which received some attention in the early 1980s, arose out of "the dilemma of the Mongol conquest in Russian historiography": embarrassment among defensive Russian nationalists who object to "Russophobic" arguments that Russia acquired "barbarian" customs, institutions, and culture from uncivilized nomads.

title: Mongol invasion of Kievan Rus'


text:
Talk:Mongol invasion of Kievan Rus'

There was no Russia back in those times
Use of term Russia is anachronism and in some places it's used even in place when it should be used term Rus. Rus is not Russia. Remember. — Preceding unsigned comment added by 195.150.224.69 (talk) 14:31, 15 February 2015 (UTC)

Really?
A significant number of historians consider the oppression of Rus' by the Mongols to be the major cause of what is sometimes called "the East-West gap" - approximately 200 years delay in introducing major social, political and economical reforms and scientific innovations in Russia compared to Western Europe. Specifically, the isolation from the West may have caused Russia's later non-involvement in the Renaissance, the Protestant Reformation, and failure to develop a middle class.

I doubt that you can call the share of historians who believe this significant. Also, the term 'delay' seems teleological and if you want to talk about this, was it not Orthodoxy with its Scripture in vernacular that cut off the Rus' lands off the Latin-speaking European countries? The Mongol invasion did cut off the Rus' from the Byzantine empire though. 83.83.1.229 (talk) 20:33, 26 November 2013 (UTC)
 * You're welcome to introduce a section on alternative theories/interpretations or a criticism section provided your information is not based on fringe theories and is well cited using verifiable secondary sources which meet the criteria for being reliable. Wikipedia is an encyclopaedic resource and adheres strictly to policies of maintaining a neutral POV and no original research. Dependent on what you qualify as being 'a significant number of historians' (sic), introducing your wishlist to the article may or may not be acceptable as we avoid giving undue weight to spurious and non-mainstream theories. That being said (and given the constraints), I'd certainly encourage you to expand the article. Cheers! --Iryna Harpy (talk) 23:11, 27 November 2013 (UTC)

The problem here is that the east-west gap is borderline a fringe theory. Countries on the border of Russia which were equally invaded by the mongols had most or part of these things. The protestant reformation failed in most of western Europe, and had a presence in areas under the eastern part of the so-called gap. The renaissance, while it didn't involve Russia, did involve most of the Mediterranean, including the ottoman empire. It sounds mostly like weird germano (this term here includes the english) centric historiography of the Weberian kind. The renaissance, for one, started in Italy, nowhere near the protestant reformation, at a time when the mongol invasions were still going on. The country where the reformation was strongest missed most of it, or even almost all of it in the case of England and Scandinavia. 216.252.76.126 (talk) 00:50, 13 February 2014 (UTC)

You do realize that the initial poster here was disagreeing with WHAT IS ALREADY in the article - the first half of their post is a quote from the article. Since consensus seems to regard that content is fringe, I'll remove it from the article.-- ¿3fam  ily6  contribs 01:41, 13 February 2014 (UTC)

Deleted content: "A significant number of historians consider the oppression of Rus' by the Mongols to be the major cause of what is sometimes called "the East-West gap" - approximately 200 years delay in introducing major social, political and economical reforms and scientific innovations in Russia compared to Western Europe. Specifically, the isolation from the West may have caused Russia's later non-involvement in the Renaissance, the Protestant Reformation, and failure to develop a middle class. "

I'm archiving the above material here.-- ¿3fam  ily6  contribs 01:41, 13 February 2014 (UTC)

 * Cheers for picking up on that, ¿3fam . I dropped the ball on the fact that the first IP was quoting from the article and thought they wanted to add spurious content. I'd certainly want to see more than one citation in order to establish that there actually are 'some' scholars. There are, in fact, a large number of factors (namely the vastness of the territory) preventing the building of a comprehensive transport infrastructure which could accommodate the needs of a burgeoning industrial sector (industrial revolution); the majority of the population being rural inhabitants and, consequently, a lesser social impact resulting from the plague/s (persistence of serfdom); etc. Mongol yoke as a major cause? Nope, never encountered that in mainstream historical analysis. --Iryna Harpy (talk) 23:33, 14 February 2014 (UTC)
 * I know that the above discussion is over a year old, but I just want to bring up that the discussion of an East-West gap looks even more absurd when you consider that Russia was one of the first countries after Italy to embrace the Renaissance.-- 3family6 ( Talk to me   |  See what I have done  ) 16:31, 12 February 2015 (UTC)

Killed in action?
Is it a correct use of word for a country? — Preceding unsigned comment added by 78.171.162.29 (talk) 14:48, 12 February 2015 (UTC)
 * No.-- 3family6 ( Talk to me   |  See what I have done  ) 16:28, 12 February 2015 (UTC)

Inconsistent terminology
In this article we have the name of the nation written as: Rus', Rusaa', Rusa', Ruas'. Let's get some consistency up in here, eh? — Preceding unsigned comment added by 70.60.254.11 (talk) 21:50, 22 January 2016 (UTC)

Requested move 24 September 2018

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: moved, per consensus. — usernamekiran (talk)  18:28, 17 October 2018 (UTC)

Mongol invasion of Rus' → Mongol invasion of Kievan Rus' – for consistency with Kievan Rus'. Marcocapelle (talk) 04:27, 24 September 2018 (UTC) --Relisting. Dreamy Jazz 🎷 talk to me | my contributions 09:01, 1 October 2018 (UTC)

 * Support - per WP:TITLE. While the invasion of Rus' occurs within the body of academic texts in a more casual sense, it is only once the subject - being the meaning of the WP:COMMONNAME "Kievan Rus'" - has been established. "Kievan Rus'", in itself, was a scholarly device ascribed the polity in the 19th and 20th centuries. The event does not refer to any events or definitions of Rus' outside of this convention. --05:01, 24 September 2018 (UTC) — Preceding unsigned comment added by Iryna Harpy (talk • contribs)
 * Support per Iryna Harpy. power~enwiki ( π, ν ) 16:31, 1 October 2018 (UTC)
 * Oppose. Kievan Rus' is commonly thought to have disintegrated ca. 1132. There was nothing Kievan about the state of Vladimir-Suzdal which bore the brunt of the invasion. --Ghirla-трёп- 08:09, 2 October 2018 (UTC)
 * Kievan Rus' is thought to have disintegrated 1240 with the Mongol invasion that the article discusses. Vladimir-Suzdal was until then one of the many states in the conglomerate ruled by the Rurik dynasty that we call Kievan Rus' nowadays. Marcocapelle (talk) 21:30, 5 October 2018 (UTC)

 * Oppose. It was the Rus' lands that were invaded not a defunct polity. Lappspira (talk) 16:16, 2 October 2018 (UTC)
 * Rus' lands has no meaning; Rus' is a redirect to Rus which is a disambiguation page. Marcocapelle (talk) 21:30, 5 October 2018 (UTC)

 * Support for Consistency in Article Titles, specifically WP:CONSUB. bd2412  T 13:42, 13 October 2018 (UTC)
 * Support. We don't have an article on Rus' in a broad sense that covered the various states after c. 1132. We use Kievan Rus' to cover its own dissolution between 1132 and 1240. I think the proposed title will therefore better serve readers. Srnec (talk) 15:01, 16 October 2018 (UTC)

 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Mongolian influence
One gets the feeling that the article was written by some stubborn Ukrainian nationalist, who considers the Russian Mongols.

1) Vasily II fought a lot with the Tatars, so blaming him for "Tatarophilia" is at least strange.

2) Most of the Tatar nobles in Russia appeared after the fall of the Golden Horde, during the conquest of the Kazan Khanate.

3) Obviously, the death penalty and torture were used only in Russia, in the rest of Europe it was not. (SARCASM)--Александр Ашкаров (talk) 03:31, 22 May 2020 (UTC)

 * Statements are sourced Vernadsky, George. (1970). The Mongols and Russia. A History of Russia, Vol. III. New Haven: Yale University Press. pp. 354-357. George Vernadsky was a Russian historian at Yale.Faustian (talk) 03:45, 22 May 2020 (UTC)
 * delete the article, if the narrow-eyed had won, they would have grazed deer in yurts 176.65.114.61 (talk) 20:30, 22 October 2022 (UTC)

tweaks
I adjusted a couple of things, one of which was the size of Batu's force - which was at least 4 tumens - not a mere 25,000 - about a third of this force (maybe even half, but that's doubtful) went to Poland after the submission of Russia, the rest to the Carpathaian basin. Garrisons had to be left in Russia. The small recon of Subetai and Jebe was two tumens (20,000) - Batu's army was much larger. I'll ref' it later. 50.111.5.65 (talk) 09:27, 31 May 2020 (UTC)

the article is clearly written with a pro-mongolian bias, and it seems to have a sinister intent
frankly, what the article seems to mirror is the position taken by the american state department, under hillary clinton. it is not likely that edits designed to correct that bias will be maintained, if the page is being actively monitored by us intelligence, as seems to be apparent from the tone.

the idea that the mongol occupation of eastern europe had some benefit to the slavs is, quite frankly, overwhelmingly offensive. it's equivalent to arguing for the upsides of the nazi occupation of poland (they got sovereignty, right?), or the benefits that africans gained from slavery (increased literacy rates, for example).

this page should be rewritten from scratch.

but, if american intelligence is monitoring it, it will revert. — Preceding unsigned comment added by 107.179.229.130 (talk) 03:16, 27 December 2021 (UTC)
 * It is way worse. Your comparisons are the wrong direction as Rus' states were more advanced than the Mongols. The Mongols did NOT bring any civilizational progress and/or knowledge to Rus' with the sole exception of new military concepts. The Mogols DID teach the (sedentary) culture Rus' people on how to wage mobile wars. About how critical it was to kill the enemy, and not bother with "holding" land. And that is about it. A better comparison would be the sacking of Rome (Kiev) and the dark ages that followed in the West (south and western Rus') with the legacy of the Roman (Russian) culture preserved in the Constantinople (Novgorod, Moscow) of the East Roman empire.145.224.105.244 (talk) 00:08, 7 July 2023 (UTC)

I suggest adding a paragraph that points to your difference in the experience of nations.
«Modern Ukraine and Belarusian lands were almost fully liberated with help of Lithuanians in 1362 on Battle of Blue Waters. A joint state was formed with the name Grand Duchy of Lithuania, Rus' and Samogitia.» Bodia1406 (talk) 23:23, 2 April 2023 (UTC)

Title
Previously this was moved from Mongol invasion of Rus' to the current title with the argument of WP:CONSISTENT. Judging by Ngram, Mongol invasion of Russia unsurprisingly is most used, but Mongol invasion of Rus' is also used more often than the current title. Thoughts? Mellk (talk) 15:25, 30 April 2023 (UTC)
 * Well, this article's current purpose it to cultivate the myth of a (centralised*) "Kievan Rus" state being separate entity from a (centralised*) "Novgorod" and a (centralised*) "Moscow" Rus. Thus as long as Ukraine exists, there is no chance factual history will show up on English WP. One just need to keep that in mind when reading anything relating Rus' and Russian history. As when reading in Soviet Pravda about happenings in the West. Just rotate 180 degrees and you are good.
 * (*) the key fake here is the deliberate treating of the Holy-Roman-Empire-like confederative supra-entities of Rus' as unitary states. That is completely fake but essential to sustain the Ukrainian narrative.145.224.105.244 (talk) 00:20, 7 July 2023 (UTC)

Curious absence of almost any mention of the most significant Kyivan Rus' successor state of the time - Galicia-Volhynia
"BY THE TIME Kyiv fell to the Mongols, it no longer reigned over others but was itself ruled by outsiders. The head of the city’s defenses, a military commander named Dmytro, owed allegiance to Prince Danylo (Daniel), ruler of Galicia and Volhynia in present-day western Ukraine. Prince Danylo had taken the Rus’ capital under his protection the previous year by arrangement with Prince Mykhailo of Kyiv, who fled after originally resisting the Mongols, then losing to them his main stronghold, the city of Chernihiv, and eventually the will to resist." - Plokhy, S. (2017). "The Gates of Europe: A History of Ukraine". Ch. 6 Pax Mongolica. (p. 49). Kobzar1917 (talk) 14:20, 28 January 2024 (UTC)

 * "The chronicler [Primary Chronicle], critical of the very idea of Christian Rus’ princes swearing allegiance to pagan Mongol khans, described three models of their behavior vis-à-vis the Mongols. Prince Mykhailo of Chernihiv exemplified the first, which met with the chronicler’s utmost approval. Since he allegedly refused Batu’s demand to kowtow before a bush and compromise his Christian religion, he was killed on orders of the khan. Prince Yaroslav of Vladimir-Suzdal represented the second model: apostasy. He allegedly agreed to bow to the bush and thereby earned the chronicler’s condemnation. Danylo followed a third model, which involved neither complete rejection of, nor full submission to, Mongol rule. According to the chronicler, who was sympathetic to Danylo, the prince did not kneel before the bush and besmirch his Christian faith, but he drank kumis, indicating acceptance of the khan’s secular authority." (Ibid., p. 53) Kobzar1917 (talk) 14:29, 28 January 2024 (UTC)
 * This should be placed in the article. Faustian (talk) 00:40, 30 January 2024 (UTC)

"Only Novgorod escaped foreign occupation."
In the introduction. Not only does it contradict the very preceding sentence and therefore make the intro unintelligible and contradictory to the rest of the article, but it's also contrary to the many sources cited. Not to mention any basic historical and geographical understanding of the Mongol conquest. I can't for the life of me imagine why an editor would keep insisting on restoring it, but here we are. Kobzar1917 (talk) 12:45, 8 July 2024 (UTC)

title: Talk:Mongol invasion of Kievan Rus'


text:
Talk:Kennedy

Etymology
What is the etymology of "Kennedy"? What does it mean? —Preceding unsigned comment added by 81.56.39.113 (talk) 11:45, 20 February 2010 (UTC)
 * Such information would be found at Kennedy (surname). Scwlong (talk) 21:07, 8 December 2012 (UTC)

Need
Does this need a separate entry? Looking for Kennedy with search should give you all the Kennedies you need. jheijmans

I cleaned up a lot of links TO this page, pointing them to the proper Kennedy articles. JimmB 22:09, 9 July 2006 (UTC)

Horses
Kennedy (horse)

Do horses get awarded Olympic medals? Andjam 06:41, 1 January 2006 (UTC)
 * Yes they do Jack Bornholm (talk) 18:58, 22 October 2012 (UTC)

Alternate spellings
Cinnéide & Chinnéide aren't alternate spellings, that's the name in Irish. (Masc/Fem) Everytime 02:16, 24 April 2007 (UTC)

John F. Kennedy in a more prominent position
I (anonymously, by mistake) moved the mention to JFK from the "See also" section to the prominent "People with the surname Kennedy". I'd guess that people looking for "Kennedy" are referring to JFK in far more cases than to ANY other entry in the disambig page. Forcing the users to scroll down so much to find his mention is insulting. More so taking into account that they have scrolled by across references to boulevards, highways, elementary/middle/high/army schools, colleges, universities, airports, bridges, space centers and stadiums named after him. Why would a casual reader want to scroll less to learn about a highway named after a former US president than about the president after whom the highway was named? — isilanes (talk|contribs) 17:16, 23 November 2007 (UTC)

 * Why is there no longer a direct link to John F. Kennedy's page? I notice there has been but then somebody removed it. There are still lots of things named after the president in that section, so why not a link to his page? I came to this page specifically looking for JFK's page, but for convenience just typed Kennedy. I think a link to his page should be readded. Princess-sarah88 (talk) 01:48, 25 March 2008 (UTC)

 * It might have been a little more convenient if you had just typed jfk (guess where it links to).--Celtus (talk) 06:38, 25 March 2008 (UTC)

 * Per WP:BEBOLD and given last mention in 2008, I went ahead and added him back. I think he's probably who we're searching for. CosineP (talk) 16:33, 31 May 2016 (UTC)

""
Disambiguation for is being discussed at talk:Kennedy family -- 65.94.171.225 (talk) 05:06, 26 September 2014 (UTC) -- 65.94.171.225 (talk) 03:31, 30 September 2014 (UTC)

title: Talk:Kennedy


text:
Dordogne

Dordogne (,  or  ;  ; Dordonha ) is a large rural department in south west France, with its prefecture in Périgueux. Located in the Nouvelle-Aquitaine region roughly half-way between the Loire Valley and the Pyrenees, it is named after the river Dordogne, which runs through it. It corresponds roughly to the ancient county of Périgord. In January 2023, Dordogne had a population of 412,807.

History
The county of Périgord dates back to when the area was inhabited by ancient celtic Gaulish tribes. It was originally home to four tribes, and since "four tribes" in the Gaulish language is "Petrocore", the area eventually became known as the county of Le Périgord. Its inhabitants became known as the Périgordins (or Périgourdins), and there are four Périgords in the Dordogne.
 * Périgord Vert (Green Périgord), with its main town of Nontron, consists of verdant valleys in a region crossed by many rivers and streams.
 * Périgord Blanc (White Périgord), situated around the department's capital of Périgueux, is a region of limestone plateaux, wide valleys, and meadows.
 * Périgord Pourpre (Purple Périgord) with its capital of Bergerac, is a wine region.
 * Périgord Noir (Black Périgord) surrounding the administrative center of Sarlat, overlooks the valleys of the Vézère and the Dordogne, and takes its name from the nearby oak and pine forests.

The Petrocores took part in the resistance against the Roman occupation of France. Concentrated in a few major sites are the vestiges of the Gallo-Roman period – the gigantic ruined tower and arenas in Périgueux (formerly Vesone), the Périgord museum's archaeological collections, villa remains in Montcaret, and the Roman tower of La Rigale Castle in Villetoureix. The earliest cluzeaux (artificial caves either above or below ground) can be found throughout the Dordogne. These subterranean refuges and lookout huts were large enough to shelter entire local populations. According to Julius Caesar, the Gauls took refuge in these caves during the resistance.

After Guienne province was transferred to the English Crown under the Plantagenets following the remarriage of Eleanor of Aquitaine in 1152, Perigord was transferred to English suzerainty. Being situated at the boundaries of influence of the monarchies of France and England, it oscillated between the two dynasties for more than three hundred years of struggle until the end of the Hundred Years' War in 1453. The county was torn apart, which modeled its physiognomy.

During the calmer periods of the late 15th and early 16th centuries, the Castillon plain on the banks of the Dordogne saw developments in urban architecture. Fine Gothic and Renaissance residences were constructed in Périgueux, Bergerac, and Sarlat. In the countryside, the nobility erected the majority of more than 1200 chateaux, manors and country houses. In the second half of the 16th century, however, as war waged in the region, attacks, pillaging, and fires of the Wars of Religion reached a pinnacle of violence, which was partly focused on Périgord. At the time, Bergerac was one of the most powerful Huguenot strongholds, along with La Rochelle. Following these wars, Périgord, fief of Henry of Navarre, was to return to the Crown for good and would continue to suffer from the various political changes of the French nation, from the Revolution to the dark days of the French Resistance.

Associated with the region are several important literary figures: Arnaut Daniel, Bertran de Born, Michel de Montaigne, Étienne de La Boétie, Brantôme, Fenelon, Maine de Biran, Eugene Le Roy, and André Maurois; as well as Talleyrand, Saint-Exupéry and Biron.

In addition to its castles, chateaux, churches, bastides, and cave fortresses, the Périgord region has preserved a number of ancient villages which retain their market halls, dovecotes, bories (stone huts), churches, abbeys, and castles. Saint-Léon-sur-Vézère, Connezac, Saint-Jean-de-Côle, La Roque-Gageac, and many others contain important and visually interesting architectural examples. The old quarters of Périgueux or Bergerac have been restored and developed into pedestrian areas. A number of small towns, such as Brantôme, Issigeac, Eymet and Mareuil, have withstood the changes of modern times. A special mention should be made in this respect to Sarlat and its Black Périgord area.

Dordogne is one of the original 83 departments created on 4 March 1790 during the French Revolution. It was created from the former province of Périgord, the county of Périgord. Its borders continued to change over subsequent decades.

 * In 1793 the communes of Boisseuilh, Coubjours, Génis, Payzac, Saint-Cyr-les-Champagnes, Saint-Mesmin, Salagnac, Savignac, Saint-Trié and Teillots were transferred from Corrèze to Dordogne.
 * In 1794 Dordogne ceded Cavarc to Lot-et-Garonne. Later in 1794 (albeit during the subsequent year under the Republican Calendar in use at the time), Dordogne gained Parcoul from Charente-Inférieure.
 * Following the restoration, in 1819, the commune of Bonrepos was suppressed and merged with the adjacent commune of Souillac in Lot.

In 1870 the region was witness to the tragic Hautefaye incident, which occurred shortly after France's declaration of war against Prussia. A young aristocrat named Alain de Monéys was brutally tortured and then burned alive, by a crowd of between 300 and 800 people on 16 August in a public square in the village of Hautefaye in the north-west region of the department. Details of the incident are gruesome, but remain unclear: the leading participants appear to have been drunk. Additionally, since the incident occurred prior to the introduction of mass education, most of the witnesses would have been unable (and possibly unwilling) to write down what they had seen. But at some stage the victim died, and following a trial four individuals identified as culpable were in turn condemned to die by guillotine. The sentence was carried out in the same public square on 13 February 1885.

It was suggested that the victim had reported some (bad) news regarding the ongoing war in a way that implied support for the enemy, although subsequently it became clear that his patriotic credentials were beyond reproach. It was also suggested that the mob had been antagonized when he called out, "Vive la République!" (Long live the republic) at a time when the patriotic villagers valued the imperial regime, which Parisian revolutionaries were in the process of destroying. This incident has been studied by historian Alain Corbin, among others.

Geography
The department is part of the region of Nouvelle-Aquitaine and is surrounded by the six départements of Haute-Vienne, Corrèze, Lot, Lot-et-Garonne, Gironde, Charente-Maritime, and Charente. Dordogne is the third-largest department of metropolitan France. In total area, it is larger than Cyprus and smaller than Cape Verde.

Principal towns
The most populous commune is Périgueux, the prefecture. As of 2019, there are 9 communes with more than 5,000 inhabitants:

Demographics
The population peaked at 505,789 in 1851 according to that year's census. After that the population declined to 373,000 by 1975. This reflected the long term population decline observed in many of the rural departments resulting from changes in agriculture and the lure of higher industrial wages available in more urbanized regions. Between 1975 and 2010, the population increased again, reaching 415,000.

Dordogne has a British immigrant community. The region has between 5,000 and 10,000 British residents and 800 British entrepreneurs, drawn by the French lifestyle, warm climate, and lower cost of living. The village of Eymet is at the heart of the trend, with 200 British families among 2,600 inhabitants.

Politics
The President of the Departmental Council is Germinal Peiro of the Socialist Party.

Tourism

 * Château de Mauriac (Douzillac)
 * Maison forte de Reignac

There are more than 1,500 castles in Dordogne, making it "The Other Chateau Country" including:

 * Beynac
 * Biron
 * Manoir de la Borie-Fricart
 * Bourdeilles
 * Castelnaud-la-Chapelle
 * Chantérac
 * Chatenet
 * Commarque
 * Hautefort
 * Manoir de Jaillac
 * Jaurias
 * La Besse
 * La Mothe
 * La Petite Filolie
 * La Renaudie
 * La Roche
 * La Roque
 * Manoir de Mitonias
 * Milandes
 * Monbazillac
 * Manoir de Grézignac
 * Manoir de la Vermondie
 * Manoir des Pautis
 * Pécany
 * Puymartin
 * Saulnier

The famous caves of Lascaux have been closed to the public, but a replica of Lascaux II is open to visitors and is a major tourist attraction. Périgueux has important Roman ruins, including an arena which is still visible inside a public park located near the town centre.

Dordogne is particularly popular with Britons as well as people from a variety of other countries, as a location for second homes.

title: Dordogne


text:
Straw man proposal

A straw-man (or straw-dog) proposal is a brainstormed simple draft proposal intended to generate discussion of its disadvantages and to spur the generation of new and better proposals. The term is considered American business jargon, but it is also encountered in engineering office culture.

Often, a straw man document will be prepared by one or two people prior to kicking off a larger project. In this way, the team can jump start their discussions with a document that is likely to contain many, but not all, of the key aspects to be discussed. As the document is revised, it may be given other edition names such as the more solid-sounding "stone-man", "iron-man", and so on.

Origins
The expression was already in use in the United States Department of Defense circa 1975 in their Large Organization Model Building paradigm (LOMB) and was apparently in use with this meaning (initial proposal) in the United States Air Force before that. The succession of names comes from the requirements document for the programming language Ada. In the High Order Language Working Group (HOLWG) the process to define Ada generated requirements documents sporting different names, representing the various stages of development of the Ada language, as described in 1993 by Col William Whitaker in an article ACM SIGPLAN Notices. They are:

 * STRAWMAN issued in April 1975
 * WOODENMAN issued in August 1975
 * TINMAN issued in January 1976
 * IRONMAN issued in January 1977 (revised in July 1977)
 * SANDMAN not published but circulated in January 1978
 * STEELMAN issued in June 1978
 * PEBBLEMAN issued in July 1978
 * PEBBLEMAN Revised and issued in January 1979
 * STONEMAN issued in February 1980

Other references include "The Ada Environment", LTC Rodney A. Edge, Walter Reed Army Institute of Research, Washington, DC 20307, Thomas A. Marciniak, M.D., National Cancer Institute, Bethesda, MD 20205

Software development
In software development, a crude plan or document may serve as the strawman or starting point in the evolution of a project. The strawman is not expected to be the last word; it is refined until a final model or document is obtained that resolves all issues concerning the scope and nature of the project. In this context, a strawman can take the form of an outline, a set of charts, a presentation, or a paper.

The ‘Straw Man’ also fits in neatly with the concept of iterative design, repeating and re-drafting to improve an initial concept or design. If anything is built, often, it may not end up in the final product. It might be that the decision is to not continue at all, in which case, the ‘Straw Man’ approach may have saved a lot of wasted work in the long run. Or, the final approach may be very different from the first model. Either way, the ‘Straw Man’ will have proven its worth by having re-directed the approach before any significant work is undertaken.

title: Straw man proposal


text:
User:Vanderesch

New Wikepedian (from 20/03/2002)

My name is Andrew van der Esch. I am of Dutch nationality but since 20 years I am living in Italy. I am a scientist working on secondary metabolites from plants. Born 1955, married and two kids.

my sandbox

title: User:Vanderesch


text:
Likelihood-ratio test

In statistics, the likelihood-ratio test is a hypothesis test that involves comparing the goodness of fit of two competing statistical models, typically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods. If the more constrained model (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.

The likelihood-ratio test, also known as Wilks test, is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test. In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent. In the case of comparing two models each of which has no unknown parameters, use of the likelihood-ratio test can be justified by the Neyman–Pearson lemma. The lemma demonstrates that the test has the highest power among all competitors.

General
Suppose that we have a statistical model with parameter space $$\Theta$$. A null hypothesis is often stated by saying that the parameter $$\theta$$ lies in a specified subset $$\Theta_0$$ of $$\Theta$$. The alternative hypothesis is thus that $$\theta$$ lies in the complement of $$\Theta_0$$, i.e. in $$\Theta ~ \backslash ~ \Theta_0$$, which is denoted by $$\Theta_0^\text{c}$$. The likelihood ratio test statistic for the null hypothesis $$H_0 \, : \, \theta \in \Theta_0$$ is given by:

 * $$\lambda_\text{LR} = -2 \ln \left[ \frac{~ \sup_{\theta \in \Theta_0} \mathcal{L}(\theta) ~}{~ \sup_{\theta \in \Theta} \mathcal{L}(\theta) ~} \right]$$

where the quantity inside the brackets is called the likelihood ratio. Here, the $$\sup$$ notation refers to the supremum. As all likelihoods are positive, and as the constrained maximum cannot exceed the unconstrained maximum, the likelihood ratio is bounded between zero and one.

Often the likelihood-ratio test statistic is expressed as a difference between the log-likelihoods
 * $$\lambda_\text{LR} = -2 \left[~ \ell( \theta_0 ) - \ell( \hat{\theta} ) ~\right]$$

where
 * $$\ell( \hat{\theta} ) \equiv \ln \left[~ \sup_{\theta \in \Theta} \mathcal{L}(\theta) ~\right]~$$

is the logarithm of the maximized likelihood function $$\mathcal{L}$$, and $$\ell(\theta_0)$$ is the maximal value in the special case that the null hypothesis is true (but not necessarily a value that maximizes $$\mathcal{L}$$ for the sampled data) and
 * $$ \theta_0 \in \Theta_0 \qquad \text{ and } \qquad \hat{\theta} \in \Theta~$$

denote the respective arguments of the maxima and the allowed ranges they're embedded in. Multiplying by −2 ensures mathematically that (by Wilks' theorem) $$\lambda_\text{LR}$$ converges asymptotically to being $χ$²-distributed if the null hypothesis happens to be true. The finite-sample distributions of likelihood-ratio statistics are generally unknown.

The likelihood-ratio test requires that the models be nested – i.e. the more complex model can be transformed into the simpler model by imposing constraints on the former's parameters. Many common test statistics are tests for nested models and can be phrased as log-likelihood ratios or approximations thereof: e.g. the Z-test, the F-test, the G-test, and Pearson's chi-squared test; for an illustration with the one-sample t-test, see below.

If the models are not nested, then instead of the likelihood-ratio test, there is a generalization of the test that can usually be used: for details, see relative likelihood.

Case of simple hypotheses
A simple-vs.-simple hypothesis test has completely specified models under both the null hypothesis and the alternative hypothesis, which for convenience are written in terms of fixed values of a notional parameter $$\theta$$:

\begin{align} H_0 &:& \theta=\theta_0 ,\\ H_1 &:& \theta=\theta_1. \end{align} $$ In this case, under either hypothesis, the distribution of the data is fully specified: there are no unknown parameters to estimate. For this case, a variant of the likelihood-ratio test is available:

\Lambda(x) = \frac{~\mathcal{L}(\theta_0\mid x) ~}{~\mathcal{L}(\theta_1\mid x) ~}. $$

Some older references may use the reciprocal of the function above as the definition. Thus, the likelihood ratio is small if the alternative model is better than the null model.

The likelihood-ratio test provides the decision rule as follows:

The values $$c$$ and $$q$$ are usually chosen to obtain a specified significance level $$\alpha$$, via the relation
 * If $$~\Lambda > c ~$$, do not reject $$H_0$$;
 * If $$~\Lambda < c ~$$, reject $$H_0$$;
 * If $$~\Lambda = c ~$$, reject $$H_0$$ with probability $$~q~$$.
 * $$~q~$$ $$ \operatorname{P}(\Lambda=c \mid H_0)~+~\operatorname{P}(\Lambda < c \mid H_0)~=~\alpha~. $$

The Neyman–Pearson lemma states that this likelihood-ratio test is the most powerful among all level $$\alpha$$ tests for this case.

Interpretation
The likelihood ratio is a function of the data $$x$$; therefore, it is a statistic, although unusual in that the statistic's value depends on a parameter, $$\theta$$. The likelihood-ratio test rejects the null hypothesis if the value of this statistic is too small. How small is too small depends on the significance level of the test, i.e. on what probability of Type I error is considered tolerable (Type I errors consist of the rejection of a null hypothesis that is true).

The numerator corresponds to the likelihood of an observed outcome under the null hypothesis. The denominator corresponds to the maximum likelihood of an observed outcome, varying parameters over the whole parameter space. The numerator of this ratio is less than the denominator; so, the likelihood ratio is between 0 and 1. Low values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared to the alternative. High values of the statistic mean that the observed outcome was nearly as likely to occur under the null hypothesis as the alternative, and so the null hypothesis cannot be rejected.

An example
The following example is adapted and abridged from.

Suppose that we have a random sample, of size $n$, from a population that is normally-distributed. Both the mean, $μ$, and the standard deviation, $σ$, of the population are unknown. We want to test whether the mean is equal to a given value, $μ0$.

Thus, our null hypothesis is $H0: μ = μ0$ and our alternative hypothesis is $H1: μ ≠ μ0$. The likelihood function is
 * $$\mathcal{L}(\mu,\sigma \mid x) = \left(2\pi\sigma^2\right)^{-n/2} \exp\left( -\sum_{i=1}^n \frac{(x_i -\mu)^2}{2\sigma^2}\right)\,.$$

With some calculation (omitted here), it can then be shown that
 * $$\lambda_{LR} = n \ln\left[ 1 + \frac{t^2}{n-1}\right] $$

where $t$ is the $t$-statistic with $n − 1$ degrees of freedom. Hence we may use the known exact distribution of $tn−1$ to draw inferences.

Asymptotic distribution: Wilks’ theorem
If the distribution of the likelihood ratio corresponding to a particular null and alternative hypothesis can be explicitly determined then it can directly be used to form decision regions (to sustain or reject the null hypothesis). In most cases, however, the exact distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine.

Assuming $H_{0}$ is true, there is a fundamental result by Samuel S. Wilks: As the sample size $$n$$ approaches $\infty$, and if the null hypothesis lies strictly within the interior of the parameter space, the test statistic $$\lambda_\text{LR}$$ defined above will be asymptotically chi-squared distributed ($$\chi^2$$) with degrees of freedom equal to the difference in dimensionality of $$\Theta$$ and $$\Theta_0$$. This implies that for a great variety of hypotheses, we can calculate the likelihood ratio $$\lambda$$ for the data and then compare the observed $$\lambda_\text{LR}$$ to the $$\chi^2$$ value corresponding to a desired statistical significance as an approximate statistical test. Other extensions exist.

title: Likelihood-ratio test