The Category of Pointed Sets .... Awodey Example 1.8, Page 17 ....

[Math Amateur]

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.6 Constructions on Categories ...

I need some further help in order to fully understand some further aspects of Example 1.8, Page 17 ... ...

Example 1.8, Page 17 ... reads as follows:View attachment 8351I find the description of the category of pointed sets confusing ... never mind the isomorphism with the coslice category mentioned ...

Can someone clarify the category of pointed sets for me ... simply speaking, what are the objects, what are the arrows, what is the identity arrow, etc Hope someone can help ...

Peter

 

[HOI]

The "category of pointed sets" is pretty much explained in the given definition: each member is a pair, a set together with a specified member of that set. If X= {a, b, c} is a set (so is in the "category of sets") then (X, a)= ({a, b, c}, a) is a member of the "category of pointed sets". (A, b) and (A, c) are other members. If Y= {p, q, r, s} is another set then (Y, p), (Y, q), (Y, r), and (Y, s) are still other members. An example or a "morphism" on this category might take (X, a) to (Y, q) or to (X, b). The "forgetful functor" from the "category of pointed sets" to the "category or sets" take each or (X, a), (X, b), and (X, c) to X and takes each of (Y, p), (Y, q), (Y, r), and (Y, s) to Y, "forgetting" the distinct point.

(A common error in learning category theory is trying to 'read too much' into the definitions. It is simpler than you think!)

 

[steenis]

What are "members" ? Do you mean objects ?

What about arrows or morphisms ? How are they defined ?

 

[Math Amateur]

What are "members" ? Do you mean objects ?

What about arrows or morphisms ? How are they defined ?

Hmmm ... thanks for the help Country Boy and Steenis ...

But I still feel a bit confused ...

I take it that the objects of \(\displaystyle \text{Sets}_\ast\) are of the form \(\displaystyle (A,a)\) and \(\displaystyle (B,b)\) where \(\displaystyle A\) and \(\displaystyle B\) are sets and \(\displaystyle a\) and \(\displaystyle b\) are specially selected elements of \(\displaystyle a,b\) respectively, called "points" ... ...

But the arrows, namely \(\displaystyle f : (A,a) \to (B,b)\), confuse me a bit because they seem to be defined in terms of the category Sets and so are not really clearly defined in \(\displaystyle \text{Sets}_\ast\) ... ...

And then there is the question of the identity arrow ...Can someone please clarify the above ..

Peter
***EDIT***

Just been reflecting on my post above ... ...

Maybe I should just relax ... :) ... and accept that a new category can be legitimately defined in terms of the objects and arrows of an established category ... ?Peter

 

[HOI]

In the category "Sets", an object is a set A or a set B and a morphism is a function, f, from set A to set B. In the category "Pointed Sets", an object is a pair, (A, p), where p is a set and p is a point in A or a pair (B, q). A morphism is a function, f, from set A to set B such that f(p)= q.

 

[Math Amateur]

In the category "Sets", an object is a set A or a set B and a morphism is a function, f, from set A to set B. In the category "Pointed Sets", an object is a pair, (A, p), where p is a set and p is a point in A or a pair (B, q). A morphism is a function, f, from set A to set B such that f(p)= q.

Thanks Country Boy ...

I appreciate all your help ...

Thanks again ...

Peter