## Franklin's Notes

### Slice categories

For any category $\mathsf{C}$, the slice category of $\mathsf{C}$ under $c$, denoted $c/\mathsf{C}$, is defined as the category whose objects are morphisms of $\mathsf{C}$ with domain $c$. The morphisms between $f:c\to x$ and $g:c\to y$ are induced by maps between the codomains $h:x\to y$, so that for each $h$ such that $g=hf$, or for which the below diagram commutes, there is a morphism between $f$ and $g$:

Specifically, corresponding to each $f:c\to x$ and $h:x\to y$ in $\mathsf{C}$, there is an element $h_f:f\to g$ of $c/\mathsf{C}$, where $g=hf$. Note that each morphism $h:x\to y$ of $\mathsf{C}$ induces several morphisms of $c/\mathsf{C}$ - exactly one for each morphism $f:c\to x$. Notice that if $h$ is a monomorphism in $\mathsf{C}$, then for each object $g$ of $c/\mathsf{C}$, there is at most one object $f$ in $c/\mathsf{C}$ such that $h_f$ has codomain $g$. Additionally, if $f$ is an epimorphism , then for any other object $g$ of $c/\mathsf{C}$, there is at most one morphism $h_f:f\to g$.

The slice category of $\mathsf{C}$ over $c$, denoted $\mathsf{C}/c$, is defined as the category whose objects are morphisms of $\mathsf{C}$ with codomain $c$. In this case, the morphisms are defined via precomposition, rather than postcomposition. That is, for each pair of morphisms $f:y\to c$ and $h:x\to y$ of $\mathsf{C}$, there is a morphism $h_f:f\to g$ of $\mathsf{C}/c$, where $g=fh$.

The "under" and "over" slice categories can be expressed in terms of each other using the opposite category construction . In particular, because we may easily define an invertible functor between $c/\mathsf{C}$ and $(\mathsf{C}^\text{op}/c)^\text{op}$ which fixes all objects and sends each morphism $f:x\to y$ to $f^\text{op op}:x\to y$.

Here are some examples:

• $1/\mathsf{Set}$ is equivalent (isomorphic, in fact) to $\mathsf{Set}^\ast$.

• $\mathbb Z[x]/\mathsf{CRing}$ is equivalent to the category of pointed (commutative) rings.

• For any category $\mathsf C$ with an object $x$, we have the isomorphism $1_x/(x/\mathsf C)\cong x/\mathsf C$. ($1_x$ is an initial object in the slice category $\mathsf C/x$.)