Understand category theory concepts in terms of an algebra of requirements.
- In my philosophy, discover a fruitful way to think about a pair (in the comma category) consisting of an object (in one category) and an arrow (to or from its analogue in a related category).
Requirements are laws. Thus I am describing how laws unfold.
Category theory concepts to describe:
- Category
- Functor
- Natural transformation
- Limits and colimits
- Left and right adjoint functors
- Exact functors
Ideas
- Consider the nature and importance of universality (for all) and uniqueness (there exists).
The Algebra of Requirements
Laws for objects
- An object has a unique arrow (the identity morphism) from itself to itself.
Laws for arrows
- An arrow goes from an object to an object. (Clarification: they may possibly be the same object but not necessarily)
Laws for categories
- Composition...
- Equivalence classes of arrows...
A notion of pairing that is weaker than identification. For different arrows may be paired with the same object, yielding different pairs.
Comma category: Given a functor {$G:\mathbf{D}\Rightarrow\mathbf{C}$} and a fixed object {$C$} in {$\mathbf{C}$}
- Pair an object {$U$} in the pre-category with an arrow {$u$} from {$C$} in the post-category.
- Pair a morphism {$\tau :U\rightarrow D$} in the pre-category with a composition {$G\tau \circ u = d \circ \textrm{id}_C$} of arrows from {$C$} in the post-category (where we leverage the identity morphism on {$C$}).
Universal mapping property
- The object (the pair) {$(U,u)$} is an initial object in the comma category {$(C\rightarrow G)$}. Thus for any other object {$(D,d)$} there is a unique morphism {$\tau_d$} in {$\mathbf{D}$} such that {$G\tau_d \circ u = d$}.
- This means that there is a unique distinction between {$U$} and any other object {$D$}, namely {$G\tau_d$}, which manifests the underlying distinction {$\tau_d$}. Note that we may have {$GU=GD$} but nevertheless {$U\neq D$} so long as {$\textrm{id}_U\neq \tau_d$}.
- The object {$(U,u)$} is known as a universal morphism from {$C$} to {$G$}.
Left adjoint functor, right adjoint functor
- A functor {$F:\mathbf{C}\Rightarrow\mathbf{D}$} is a left adjoint functor if for each object {$X$} in {$\mathbf{C}$} there exists a universal morphism {$(G(X),g_X)$} from {$F$} to {$X$}.
- A functor {$G:\mathbf{D}\Rightarrow\mathbf{C}$} is a right adjoint functor if for each object {$Y$} in {$\mathbf{D}$} there exists a universal morphism {$(F(Y),f_Y)$} from {$Y$} to {$G$}.