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

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In summary, the category of pointed sets is a subcategory of the category of sets where objects are pairs consisting of a set and a specified element from that set, and arrows are defined as functions between these sets that preserve the specified elements. The identity arrow in this category is a function that maps the specified element of an object to itself.
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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
 
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  • #2
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!)
 
  • #3
What are "members" ? Do you mean objects ?

What about arrows or morphisms ? How are they defined ?
 
  • #4
steenis said:
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
 
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  • #5
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.
 
  • #6
Country Boy said:
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
 

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

1. What is the Category of Pointed Sets?

The Category of Pointed Sets is a mathematical concept that represents a category containing objects and arrows between them. The objects are sets with a chosen point, and the arrows are functions that preserve the chosen point.

2. What is the significance of Awodey Example 1.8?

Awodey Example 1.8 is a specific example given by philosopher and mathematician Steve Awodey in his book "Category Theory" to illustrate the concept of the Category of Pointed Sets. It helps to understand the definition and properties of this category.

3. How is the Category of Pointed Sets different from other categories?

The Category of Pointed Sets is different from other categories because it has an additional structure of a chosen point within each set. This point has to be preserved by the arrows in order to maintain the category structure.

4. What is the purpose of studying the Category of Pointed Sets?

Studying the Category of Pointed Sets can help in understanding the fundamentals of category theory and its applications in various fields such as mathematics, computer science, and physics. It also provides a framework for reasoning about structures with a chosen element.

5. Are there any real-world applications of the Category of Pointed Sets?

Yes, the Category of Pointed Sets has applications in computer science, particularly in programming languages and computer graphics. It also has connections to topology and algebraic geometry, where the chosen point can represent a base point for a space or an algebraic structure.

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