\(\def\C{ \mathbf{C} }\) \(\def\cat#1{ \mathbf{#1} }\) \(\def\domain#1{ \mathrm{dom}(#1) }\) \(\def\o{\circ}\) \(\def\cmp{\circ}\)

1.7 CATEGORIES: PRODUCTS, COPRODUCTS, AND FREE OBJECT

• Several of mathematical objects A already introduced (sets, groups, monoids) (rings, modules) with the appropriate maps of these objects (set:functions, groups:homomorphisms) have a number of formal properties in common.

• For example, composition of maps is associative; each object A has an identity map \(1_A :A - > A\) with certain properties.

• DEF Category is a class \(\cat{C}\) of objects (A,B,C) with

  • (i) class of disjoint sets, \(hom(A,B)\), one for each pair of objects in \(\cat{C}\)
    • morphism: is \(f \in hom(A,B)\) \(f:A- > B\)

  • (ii) for each triple \((A,B,C)\) from \(\cat{C}\) a function

    • composition of morphism: \(hom(B,C) \times hom(A,B) - > hom(A,C)\)
      • \(f:A- > B\), \(g:B- > C\) \(g \circ f\)
    • subject to axioms
    • Associative: for \(f,g, h:C- > D\)
      • \(h \o ( g \o f ) = (h \o g) \o f \)
    • Identity: for each B of \(\cat{C}\) there exists morphism \(1_B:B- > B\) s.t.
      • for any \(f:A- > B g:B- > A\),
        • \(1_B \cmp f = f\) and \(g \cmp 1_B = g\)

  • NOTES:

    • The morphism defines some relationship that must be preserved transitively
    • The morphism also forms the basis for defining equivalence

• DEF a morphism \(f\) is equivalence if there is \(g\) such that \(g \cmp f=1_A\) and \(f \cmp g=1_B\)

  • the composite of two equivalences, when defined, is an equivalence
  • if \(f:A- > B\) is an equivalence, then \(A\) and \(B\) are equivalent

• Let's carefully think about what \(1_A\) means

  • is it the same as an atypical identity? or are there some restrictions?

• these symbols are kind of frustrating me haha

• Why is the ordering of the hom(A,B) and but the composition is written the other way?

  • hom read right, composition read left

• DEF Product: Given a family of objects \(\{A_i | i\in I\}\) a product, \(P\), is an object with \(\pi_i : P \rightarrow A_i\)

  • so that for any other object, \(B\), with mappings to the family \(\psi_i: B \rightarrow A_i\)
  • there is unique \(f: B \rightarrow P\) with \(\pi_i \circ f = \psi_i\)
  • \(P \rightarrow A_i \leftarrow B\)
  • \(f:B \rightarrow P\)

• THM two product of the same family are equivalent

• consider two products P,B and \(f:B \rightarrow A\)
• we need to show \(f \circ g = 1\) and \(g \circ f = 1\)
  • \(f \circ g\) and \(1\) both satisfy link function of commuting the family mapping from P to P
    • since the link function is unique, they are equal. This means f is inverse of g
    • so the two products are equivalent
• draw out the diagram, it is quite interesting how you can put the two \(P \rightarrow A_i \leftarrow B\) maps together to form
• ??? What does it mean for comm diagram to commute?

• DEF Coproduct/Sum: Same definition as Product except the mappings to family and link mapping are reversed
  • \(S \leftarrow A_i \rightarrow B\)
  • with unique \(f:S \rightarrow B\)
  • examples:
    • The direct sum definition in linear algebra says that the sum must have unique representation or sum of vector subspaces whose intersection is zero
• DEF F is Free on X if
  • \(F \leftarrow X \rightarrow B\)
  • with unique \(f:S \rightarrow B\)
• THM two co products of same family are equivalent
• DEF initial element: Let \(I\) be an initial element, then for any other object \(O\), there is unique morphism from \(I \rightarrow O\)
• DEF terminal element

• initial/terminal property is what you need to say any two initial/terminal objects are equivalent

• EXAMPLE Deconstruction of Free object into an initial object

  • The morphism from \(f:F \rightarrow O\) such that \(f \circ i = j\) is unique
  • So, can we build a category where I and O where f is the mapping between them?
  • NOTE: I like this exercise a building something that will fit a certain defintion/property
    • I like it becaause its making the more complex space simpler (a morphism in our new space means the commute relationship must hold automatically)
• EXAMPLE Deconstruction of Product into an initial object
• Can you build something similar for Products and Coproducts

Exercises

• ??? Are there any interesting exercises in particular? that illustrates an interesting point?
• DEF pointed sets
• 1. Inverse of equivalent morphism is unique
• 1. Deriving a unique morphism for product of group
  2. we describe f that commutes \(\pi_i \circ f = f_i\)
  3. is f a morphism of groups? is it unique?
• 1. Same as #3 but for coproduct of group
• 1. Construct a coproduct of sets
• 1. Construct a
  2. product for pointed sets
  3. coproduct for pointed sets
• 1. The properties of a Free object can force i to be injective because it is linked to the other objects
• 1. In category of groups, \(F\) is isomorphic to \(G\) due to identity morphism working as a unique linking map between \(F\) and \(G\)
  2. \(F\) is free with \(X\), with map \(i:X \rightarrow F\)
  3. let \(G\) be group generated by i(X)

Questions To Teach

• What is the definition of category
  • what is the homomorphisms
  • transitive
  • equivalence
  • identity?
  • What is the point of this defintion, what is it trying to capture?
• ??? What are products in a category?
  • Could you find an example of one for sets?
  • What is coproduct
  • What are free objects?
  • What are initial/terminal objects?
  • What is the relationship between product/coproduct/free/initial/terminal objects
  • Is there relationship between a initial object and a terminal object