Topics to be covered.
- Definition of Categories
- Some history and motivation
- Some example Categories: Category of Sets, Posets, Sets with Structure, Deductions systems.
- Constructions on Categories: Products, arrow categories, opposite category.
- Foundations, questions of size, the category of categories? and how to stop your head hurting.
Sections of the book
- 1.1-1.6 and 1.8
Exercises to be Done.
- Exercises 1-8, skip exercise 4 if you know nothing about topology, unless you are interested in learning something then look up the definition of a topological space. As a bonus exercise 14 is quite fun.
Hint for exercise 2 b
What is the category of sets? The objects are sets, but what is an arrow between sets? If you look up a set-theory textbook you’ll find the following definition of a function. A function $f$ between two sets $A$ and $B$ is a subset of $A\times B$, that is $f \subseteq A \times B$, such that:
- For all elements $a \in A$ there exists some pair $(a,b) \in f$
- $(a,b) \in f$ and $(a,b’) \in f$ implies that $b=b’$.
We say that $f(a) = b$ if $(a,b) \in f$.
Condition $1$ says that all the elements in $A$ are mapped to something in $B$, and condition $2$ makes it act like a function. $f(a)$ is a single value.
Think about these two questions: Given some set $A$
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How many functions are from the empty set $\emptyset$ to $A$?
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How many functions are there from $A$ to $\emptyset$.
To answer this question you have to ask yourself what $A\times \emptyset$ and what $\emptyset \times A$ equals.