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Page "Groupoid" ¶ 60
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unique and functor
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.
which is natural in the variable N. Here the functor Hom ( N, F –) is the composition of the Hom functor Hom ( N, –) with F. This isomorphism is the unique one which respects the limiting cones.
For a given diagram F: J C and functor G: C D, if both F and GF have specified limits there is a unique canonical morphism
A functor G lifts limits uniquely for a diagram F if there is a unique preimage cone ( L ′, φ ′) such that ( L ′, φ ′) is a limit of F and G ( L ′, φ ′) =
) The variety that represents this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists.
: A universal element of a functor F: C Set is a pair ( A, u ) consisting of an object A of C and an element u ∈ F ( A ) such that for every pair ( X, v ) with v ∈ F ( X ) there exists a unique morphism f: A X such that ( Ff ) u = v.
* The forgetful functor U: Grp Set is faithful as each group maps to a unique set and the group homomorphism are a subset of the functions.
This is enough to show that right derived functors of any left exact functor exist and are unique up to canonical isomorphism.
A functor F: 1 Set maps the unique object of 1 to some set S and the unique identity arrow of 1 to the identity function 1 < sub > S </ sub > on S. A subfunctor G of F maps the unique object of 1 to a subset T of S and maps the unique identity arrow to the identity function 1 < sub > T </ sub > on T. Notice that 1 < sub > T </ sub > is the restriction of 1 < sub > S </ sub > to T. Consequently, subfunctors of F correspond to subsets of S.
Formally, the right Kan extension of along consists of a functor and a natural transformation which is couniversal with respect to the specification, in the sense that for any functor and natural transformation, a unique natural transformation is defined and fits into a commutative diagram
This gives rise to the alternate description: the left Kan extension of along consists of a functor and a natural transformation which are universal with respect to this specification, in the sense that for any other functor and natural transformation, a unique natural transformation exists and fits into a commutative diagram:
where is the unique functor from to

unique and F
Namely φ is universal for homomorphisms from G to an abelian group H: for any abelian group H and homomorphism of groups f: G H there exists a unique homomorphism F: G < sup > ab </ sup > H such that.
However, in principle, since the same electronegativities should be obtained for any two bonding compounds, the data is in fact overdetermined, and the signs are unique once a reference point is fixed ( usually, for H or F ).
If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra over F there is a simply connected Lie group G with as Lie algebra, unique up to isomorphism.
If F is a field and f and g are polynomials in F with g ≠ 0, then there exist unique polynomials q and r in F with
As Mason uses his unique experience to escape from their cells, he reveals why he was held there for so many years — for stealing a microfilm of the United States ' most closely guarded secrets, including the Roswell UFO incident and the John F. Kennedy assassination ( Womack revealed this to Paxton, earlier ).
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
As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is that given a set g < sub > 1 </ sub >, and class functions G < sub > 2 </ sub >, G < sub > 3 </ sub >, there exists a unique function F: Ord V such that
* For each object X in C, ( F ( X ), η < sub > X </ sub >) is an initial morphism from X to G. That is, for all f: X G ( Y ) there exists a unique g: F ( X ) Y for which the following diagrams commute.
That is, for all g: F ( X ) Y there exists a unique f: X G ( Y ) for which the following diagrams commute.
A limit of the diagram F: J C is a cone ( L, φ ) to F such that for any other cone ( N, ψ ) to F there exists a unique morphism u: N L such that φ < sub > X </ sub > o u =
A colimit of a diagram F: J C is a co-cone ( L, ) of F such that for any other co-cone ( N, ψ ) of F there exists a unique morphism u: L N such that u o < sub > X </ sub > = ψ < sub > X </ sub > for all X in J.
As with limits, if a diagram F has a colimit then this colimit is unique up to a unique isomorphism.
which assigns each diagram its limit and each natural transformation η: F G the unique morphism lim η: lim F lim G commuting with the corresponding universal cones.

unique and
If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K H such that f = h φ.
If it does, however, it is unique in a strong sense: given any other inverse limit X ′ there exists a unique isomorphism X ′ X commuting with the projection maps.
The homomorphism η is characterized by the following universal property: given any profinite group H and any group homomorphism f: G H, there exists a unique continuous group homomorphism g: G < sup >^</ sup > H with f
* Whenever Y is an object of D and f: X U ( Y ) is a morphism in C, then there exists a unique morphism g: A Y such that the following diagram commutes:
* Whenever Y is an object of D and f: U ( Y ) X is a morphism in C, then there exists a unique morphism g: Y A such that the following diagram commutes:
such that for any other object Z of D and morphisms f: Z X and g: Z Y there exists a unique morphism h: Z X × Y such that f
Specifically, it is unique up to a unique isomorphism: if ( A ′, φ ′) is another such pair, then there exists a unique isomorphism k: A A ′ such that φ ′
Suppose ( A < sub > 1 </ sub >, φ < sub > 1 </ sub >) is an initial morphism from X < sub > 1 </ sub > to U and ( A < sub > 2 </ sub >, φ < sub > 2 </ sub >) is an initial morphism from X < sub > 2 </ sub > to U. By the initial property, given any morphism h: X < sub > 1 </ sub > X < sub > 2 </ sub > there exists a unique morphism g: A < sub > 1 </ sub > A < sub > 2 </ sub > such that the following diagram commutes:

unique and is
What makes the current phenomenon unique is that so many science-fiction writers have reversed a trend and turned to writing works critical of the impact of science and technology on human life.
One of the inescapable realities of the Cold War is that it has thrust upon the West a wholly new and historically unique set of moral dilemmas.
But Oakwood Heights is unique in one particular.
The structure appears to be unique among OOH compounds, but is the same as that assumed by Af.
A number of unique medical problems might be created when man is exposed to an infectious agent through the respiratory route rather than by natural portal of entry.
These operators D and N are unique and each is a polynomial in T.
It is interesting that a 1: 1 correspondence can be established between the lines of two such pencils, so that in a sense a unique image can actually be assigned to each tangent.
Hence, thought of as a line in a particular plane **yp, any tangent to Q has a unique image and moreover this image is the same for all planes through L.
Spontaneity training theory is unique and relatively new.
This weakness is not unique to labor surplus areas, for it is inherent in the system of local school districts in this country.
What with traders trading for so many different objectives, and what with there being so many unique and individualized market theories and trading techniques in use, and more coming into use all the time, it is hard to imagine how any particular theory or technique could acquire enough `` fans '' to invalidate itself.
Probably the primary reason for special treatment of a net operating loss carryover is the unique opportunity it presents for tax avoidance.
It is the classroom teacher, however, who has daily contacts with pupils, and who is in a unique position to put sound psychological principles into practice.
One reason for the unique vitality of the chorus is its great variety in expression.
The policy may not be unique but the maximum value of P certainly is, and once the policy is specified this maximum can be calculated by ( 2 ) and ( 3 ) as a function of the feed state Af.
Sir Julian Huxley in his book Uniqueness Of Man makes the novel point that just as man is unique in being the only animal which requires a long period of infancy and childhood under family protection, so is he the only animal who has a long period after the decline of his procreativity.
Most people do not realize that the congregation, as a gathered fellowship meeting regularly face to face, personally sharing in a common experience and expressing that experience in daily relationships with one another, is unique.
A sense of self-certainty and the freedom to experiment with different roles, or confidence in one's own unique behavior as an alternative to peer-group conformity, is more easily developed during adolescence if, during early childhood, the individual was permitted to exercise initiative and encouraged to develop some autonomy.
First, the State Department is unique among government agencies for its lack of public supporters.
The death of a man is unique, and yet it is universal.

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