Terminal object

Hierarchical methodology was chosen for the programme Structure of an Atom. There is an ordered relationship between sub-skills which ultimately help to learn the terminal objective. The simple things are given first that makes the basis for more complex concepts. 

The frames were constructed in the same sequence as shown in the task analysis. The frames were sequenced in such a way that by completing all the frames the terminal objective might be achieved by the students. 

Figure 7.7 The three-dimensional Van de Vusse system for different terminating objective functions (a) Cp = 0.3 mol/L and (b)...

A terminal object of a category C is an object term G 0(C) such that for any object a G 0 C), there exists a unique morphism a term. 

Initial and terminal objects do not have to exist. In any case, the category can be extended by adding an element with a unique morphism to or from every other object. Even if the initial and terminal objects already exist, they are not necessarily unique. The good news, as the next proposition shows, is that they are unique up to isomorphism. 

Proposition 4.17. In an arbitrary category C, any two initial objects are isomorphic, and any two terminal objects are isomorphic. 

Proof. We give an argument for two initial objects. To obtain the proof for terminal objects, simply reverse all the arrows. Let initi and init2 be initial objects of C. Since initi is an initial object, there exists a unique morphism a initi — init2. Since init2 is an initial object, there exists a unique morphism / init2 —> initi. Since there exists only one morphism from initi to itself, we conclude that the composition morphism / oa must be an identity. In the same way a o / must be an identity. This proves that a and / are isomorphisms, and hence initi and init2 are isomorphic. ... 

Initial and terminal objects, as well as products and coproducts, are special cases of more general constructions of limits and colimits. The latter are defined and investigated in Section 4.4. 

The category Cat also has terminal objects these are categories with one object, whose only automorphism is the identity. 

Examples of limits are provided by terminal objects, as well as byproducts. 

We note that for an arbitrary acyclic category C, the acyclic category C 1 is obtained from C by adding a terminal object, while the category 1 C is obtained from C by adding an initial one. In the special case of posets we recover the following classical concept. 

Proof. We observe that since t is a terminal object, for any mi G M C) satisfying d m = d mi, there exists a unique morphism m2 such that niiom = m. In particular, we have an object (mi, m2) of Im, and the second morphism in this pair is uniquely determined by the first. We recall Definition 4.35 and see that this fact gives a bijection between objects of Im and objects of Clower triangle is always trivially satisfied. This leaves us with requiring just the commutativity of the upper triangle, which under the bijection above translates exactly to the commutativity condition in Definition 4.35. This shows that the above bijection extends to yield an isomorphism of categories. ... 

If both C and D have initial objects sc and sd, as well as terminal objects tc andto, then... 

Theorem 10.26. For any finite acyclic category C with an initial object s and a terminal object t, we have... 

So let us assume that G is a finite acyclic category, and let C denote the category obtained from C by augmenting it with an initial and a terminal object. We would like to shell the nerve A(C). This time around, we shall label not the edges in the Hasse diagram of a poset, but rather the morphisms that cannot be represented as a composition of two morphisms, none of which is an identity. Recall that we called such morphisms indecomposable. 

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