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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.
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colimit and diagram
As with limits, if a diagram F has a colimit then this colimit is unique up to a unique isomorphism.
Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.
A given diagram F: J → C may or may not have a limit ( or colimit ) in C. Indeed, there may not even be a cone to F, let alone a universal cone.
A category has colimits of type J if every diagram of type J has a colimit in C. A cocomplete category is one that has all small colimits.
Dually, if every diagram of type J has a colimit in C ( for J small ) there exists a colimit functor
which assigns each diagram its colimit.
Dually, if a diagram F: J → C has a colimit in C, denoted colim F, there is a unique canonical isomorphism
Identifying the limit of Hom ( F –, N ) with the set Cocone ( F, N ), this relationship can be used to define the colimit of the diagram F as a representation of the functor Cocone ( F, –).
* In category theory, a category C is complete if every diagram from a small category to C has a limit ; it is cocomplete if every such functor has a colimit.
A coequalizer is a colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g: X → Y.
In category theory, a branch of mathematics, a pushout ( also called a fibered coproduct or fibered sum or cocartesian square or amalgamed sum ) is the colimit of a diagram consisting of two morphisms f: Z → X and g: Z → Y with a common domain: it is the colimit of the span.
between the colimit functor that assigns to any diagram in C indexed by some category I its colimit and the diagonal functor that maps any object c of C to the constant diagramm which has c at all places.
colimit and F
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.
* A colimit of F is a universal morphism from F to Δ.
For instance, a functor G preserves the colimits of F if G ( L, φ ) is a colimit of GF whenever ( L, φ ) is a colimit of F. A cocontinuous functor is one that preserves all small colimits.
colimit and →
* the functor H: I → C has limit ( or colimit ) l if and only if the functor FH: I → D has limit ( or colimit ) Fl.
colimit and is
The central concept which is needed for this purpose is called categorical limit, and can be dualized to yield the notion of a colimit.
Similar remarks apply to the colimit functor ( which is covariant ).
In mathematics, a direct limit ( also called inductive limit ) is a colimit of a " directed family of objects ".
colimit and such
The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.
colimit and morphism
The notion of a universal morphism to a particular colimit, or from a limit, can be expressed in terms of a comma category.
colimit and X
* The stable cohomotopy group of X is the colimit
The following isomorphism shows that a simplicial set X is a colimit of its simplices:
where the colimit is taken over the simplex category of X.
where the colimit is taken over the n-simplex category of X.
In general topology and related areas of mathematics, the final topology ( or strong topology or colimit topology or inductive topology ) on a set, with respect to a family of functions into, is the finest topology on X which makes those functions continuous.
colimit and .
The concepts of limit and colimit generalize several of the above.
( Section 2. 7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids ).
Note that both the limit and the colimit functors are covariant functors.
diagram and F
If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:
* F transforms each commutative diagram in C into a commutative diagram in D ;
A, B, C, D, E, F, Z, U, V, G, T, H, M, L, K, I, J on diagram as head and tail, A-F code for phage head genes, Z-J code for phage tail genes.
The category J is thought of as index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J.
Let F: J → C be a diagram of type J in a category C. A cone to F is an object N of C together with a family ψ < sub > X </ sub >: N → F ( X ) of morphisms indexed by the objects of J, such that for every morphism f: X → Y in J, we have F ( f ) o ψ < sub > X </ sub >
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 co-cone of a diagram F: J → C is an object N of C together with a family of morphisms
In the following we will consider the limit ( L, φ ) of a diagram F: J → C.
A cone to the empty diagram is essentially just an object of C. The limit of F is any object that is uniquely factored through by every other object.
If J is a discrete category then a diagram F is essentially nothing but a family of objects of C, indexed by J.
A special case of a product is when the diagram F is a constant functor to an object X of C. The limit of this diagram is called the J < sup > th </ sup > power of X and denoted X < sup > J </ sup >.
diagram and J
Diagram: For categories C and J, a diagram of type J in C is a covariant functor.
Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of type J in C is a functor from J to C:
A diagram is said to be small or finite whenever J is.
If J is the empty category there is only one diagram of type J: the empty one ( similar to the empty function in set theory ).
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