Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.

Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

* Given any morphism k ′: K ′ → X such that f k ′ is the zero morphism, there is a unique morphism u: K ′ → K such that k u

Given two reductive groups and a ( well behaved ) morphism between their corresponding L-groups, this conjecture relates their automorphic representations in a way that is compatible with their L-functions.

Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as

Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category C < sup > op </ sup >.

* Given any manifold, there is a Lie groupoid called the pair groupoid, with as the manifold of objects, and precisely one morphism from any object to any other.

* Given a Lie group acting on a manifold, there is a Lie groupoid called the translation groupoid with one morphism for each triple with.

Given a category C and a morphism

Given any morphism between objects X and Y, if there is an inclusion map into the domain, then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain R → Y known as the range of f.

Given and, a morphism in the coslice category is a map making the following diagram commute:

Given two representations ρ: G → GL ( n, C ) and τ: G → GL ( m, C ) a morphism between ρ and τ is a linear map T: C < sup > n </ sup > → C < sup > m </ sup > so that for all g in G we have the following commuting relation: T ° ρ ( g ) = τ ( g ) ° T.

Given a morphism, any vector bundle on Y, or more generally any sheaf in modules, eg.

#( TR 3 ) Given a map between two morphisms, there is a morphism between their mapping cones ( which exist by axiom ( TR 1 )), that makes everything commute.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given a function f of type, currying it makes a function.

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).

Given the laws of exponents, f ( x )

Given a function f of a real variable x and an interval of the real line, the definite integral

Given a function ƒ defined over the reals x, and its derivative ƒ < nowiki > '</ nowiki >, we begin with a first guess x < sub > 0 </ sub > for a root of the function f. Provided the function is reasonably well-behaved a better approximation x < sub > 1 </ sub > is

# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

Given f

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system

: Given any set A and any set B, if for every set C, C is a member of A if and only if C is a member of B, then A is equal to B.

: Given any set A and any set B, if A is a nonempty set ( that is, if there exists a member C of A ), then if A and B have precisely the same members, then they are equal.

: Given any set A and any set B, there is a set C such that, given any set D, D is a member of C if and only if D is equal to A or D is equal to B.

: Given any set A, there is a set B such that, given any set x, x is a member of B if and only if x is a member of A and φ holds for x.

: Given any set A, there is a set such that, given any set B, B is a member of if and only if B is a subset of A.

: Given any set A, there is a set B such that, for any element c, c is a member of B if and only if there is a set D such that c is a member of D and D is a member of A.

Given a circle A, find a circle B such that the area of the intersection of A and B is equal to the area of the symmetric difference of A and B ( the sum of the area of A − B and the area of B − A ).

Given a core geometry, the B field needed for a given force can be calculated from ( 2 ); if it comes out to much more than 1. 6 T, a larger core must be used.

Given two candidates, A and B, A is better than B on a constraint if A incurs fewer violations than B.

With regard to African Americans, Gilman wrote in the American Journal of Sociology: “ The problem, is this: Given: in the same country, Race A, progressed in social evolution, say, to Status 10 ; and Race B, progressed in social evolution, say, to Status 4..

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

Given a class function G: V → V, there exists a unique transfinite sequence F: Ord → V ( where Ord is the class of all ordinals ) such that

Given a function f: X → Y, the set X is the domain of f ; the set Y is the codomain of f. In the expression f ( x ), x is the argument and f ( x ) is the value.

* Given a category C with finite coproducts, a cogroup object is an object G of C together with a " comultiplication " m: G → G G, a " coidentity " e: G → 0, and a " coinversion " inv: G → G, which satisfy the dual versions of the axioms for group objects.

Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).

Given a vector x ∈ V and y * ∈ W *, then the tensor product y * ⊗ x corresponds to the map A: W → V given by

: Given two sets, A and T, of equal size, together with a weight function C: A × T → R. Find a bijection f: A → T such that the cost function:

Given a concrete category ( C, U ) and a cardinal number N, let U < sup > N </ sup > be the functor C → Set determined by U < sup > N </ sup >( c ) = ( U ( c ))< sup > N </ sup >.