So to basically summarize the pattern that we have been using: we introduced mathematical objects, then some generalized notion of mappings between objects which we called "morphisms". We then collected a bunch of objects and morphisms together such that a bunch of nice properties were true, and we called it a category.

A category "is-a" mathematical object, so we figured out what morphisms between categories were: functors, or mathematical procedures. We then said "Hey, a functor 'is-a' mathematical object, what's a morphism between them?" It turns out it's a natural transformation.

But a morphism is-a mathematical object. So what's a "morphism" between morphisms? Is this a "meaningful" question? (It is, but first some references!)

- Saunders Mac Lane,
*Categories for the Working Mathematician*, Chapter II.6 "Comma Categories" - The Catsters, Youtube Lecture Series "Slice and Comma Categories", 1, 2

(We hope that the reader can see the pattern here, in category theory we started out with remarkably simple, almost comical, premises. We have objects, and we have morphisms. We group all objects of the same sort together into a category. But we can apply this powerful notion of a morphism to any object, and everything in math "is-a" mathematical object. We can then ask, for each object, what's the morphism of this thing? What's a morphism of this morphism? And so on...)

### Category of Objects Under and Over a Specific Object

Now, to discuss morphisms between objects, we need to demand that they are the "same sort" of objects. How to do this? Well, we can demand that the morphisms are from one fixed object or dually they go to one fixed object.

More precisely, let *b*∈**C** be a fixed object, we can construct "*the category of objects under $b$*" denoted by $(b\downarrow\mathbf{C})$. Basically the objects are ordered pairs $\langle f,c\rangle$ consisting of morphisms from $b$ and the target of the morphism, and the morphisms are morphisms of the target $h:\langle f,c\rangle\to\langle f',c'\rangle$...well, this doodle should summarize things nicely:

Composition of morphisms is given by composition of these triangles. More precisely, the composition of the morphisms in the base of the triangle. To be formal, lets give the full definition:

Definition 1.LetCbe a category,b∈Cbe some element. Then a"category of objects underconsists ofb"such that the following diagram specifies this

- (Objects) the collection of ordered pairs ⟨f,
c⟩ where $f:b\to{c}$ is a morphism inC, andc∈Cis an object;- (Morphisms) the morphisms are
h: ⟨f,c⟩ → ⟨f',c'⟩

Consider any one point set $*$, and any other set $X$. The function $*\to{X}$ is a single element of $X$. The category of objects under $*$ consist of objects $\langle *\to{X},X\rangle$ which is a selected element of $X$, and the set $X$. This is precisely **Set _{*}**, the first part of the ordered pair (the morphism) is the "base point", the second part of the ordered pair is the set itself.

Now, as alluded to earlier, we can reverse the direction of the arrows by "duality". This ends up letting us consider morphisms with the same target.

That is, if we have a category **C** and an object *a*∈**C**, then we can construct a category with the objects being ⟨*c*,f⟩ where f : *c* → *a* is a morphism.

The morphisms in this category would be "morphisms of morphisms". More precisely, it would make the following diagram (on the right) commute:

As we reversed the direction of the arrows, we should probably distinguish this sort of category of objects over *a* from the category of objects under *b*. We denote it by (**C** ↓ *a*), compared to (*b* ↓ **C**) for objects under *b*.

Formally, this is summarized in the following definition:

Definition 2.LetCbe a category,a∈Cbe some element. Then a"category of objects overdenoted by (a"C↓a) consists ofsuch that the following summarizes the category:

- (Objects) the collection of ordered pairs ⟨f,
c⟩ where $f:c\to{a}$ is a morphism inC, andc∈Cis an object;- (Morphisms) the morphisms are
h: ⟨f,c⟩ → ⟨f',c'⟩

Observe that this is just the same as the category of objects under *b*, except we have the arrows be reversed.

We should consider, to be kosher, what (**Set** ↓ ***) is. The objects are ordered pairs, as before, consisting of ⟨*X* → *,*⟩ where the first component of the ordered pair is unique. There is always a map in **Set** from *X* to a singleton ***, it's very boring though -- it's the constant map. It's a unique map too (this is kind of trivial, where else could it map stuff to? There's only one choice!).

The morphisms are really just morphisms between the sets.

Lets think for a second: for each object *X*∈**Set** there is a unique object ⟨*X* → *,*⟩ in (**Set** ↓ ***); and for each morphism *f*∈hom_{Set}(*X*,*Y*) there is a corresponding morphism in (**Set** ↓ ***).

Since there is a unique one-to-one correspondence between objects and between morphisms in the two categories, we can represent this by a functor which is invertible. What does this mean? It's an isomorphism...i.e. (**Set** ↓ ***) is isomorphic to **Set**!!!

#### Remark: Notation

Apparently Mac Lane uses two notations. One is what we've seen. The other is writing **C**/c for (**C** ↓ c), and (one assumes) c/**C** for (c ↓ **C**). Just be warned there are two different notations (apparently).

[More to follow...apologies to those who would've liked more, duty calls at my job]