Category

We first define ‘category’, which can be interpreted as a system of a specific mathematical concept. The category consists of ‘objects’ and ‘morphisms’ with ‘composition’, which satisfies the axioms of a monoid. This makes a naturality of composing several morphisms. From the category $\mathcal{A}$, we have those notations:

• $\text{ob}(\mathcal{A})$ - collection of all objects
• $\mathcal{A}(A,B)$ - collection of all morphisms from A to B(is called a homset if it is a set, but what is set? We will deal with it in section 3.)
• $1_A$ - identity morphism from A to A

We also have definitions of ‘isomorphism’, ‘inverse’, and ‘isomorphic’, which are quite straightforward.

We can construct new categories such as $\mathcal{A}^{op},\,\mathcal{A}\times\mathcal{B}$, which are dual and product, respectively. The concept of the dual is essential as it can not only make the proof in category theory in half but also make a strong link between the theory and its dual theory.

Functor

Next is about ‘functor’, denoted as $\rightarrow$, which looks like a morphism between categories(actually, it is really a morphism in the category of category, which we define as $\textbf{CAT}$). It sends all objects and morphisms from the former one to the latter one, satisfying a homomorphism-like axiom. This makes a naturality between functor and composition.

There are some important functors that can be observed in many mathematical theories; free functor and forgetful functor. For example in group theory, a free functor sends a set to its free group.

For category $\mathcal{A}$ and $\mathcal{B}$, we distinguish some kinds of functors:

• $F:\mathcal{A}\rightarrow\mathcal{B}$ - covariant functor
• $F:\mathcal{A}^{op}\rightarrow\mathcal{B}$ - contravariant functor
• $F:\mathcal{A}^{op}\rightarrow\textbf{Set}$ - presheaf on $\mathcal{A}$

The contravariant functor usually arises when we assign objects to morphism space whose element goes to some fixed object.

We also define a property for some functors:

• faithful - morphisms go injective,
• full - morphisms go surjective.

Note: Don’t think of the image of a functor as a subcategory; it doesn’t satisfy a condition for the category in many cases.

Natural Transformation