Math and science::Algebra::Aluffi

Slice category

A slice category is an example of a category whose objects are morphisms of some other category and whose morphisms are also morphism of that original category.

The objects of a slice category are ambient morphisms to an object in an ambient category, and the morphisms of a slice category are ambient morphisms from one slice category object to another. The precise definition is as follows:

Slice category

Let $C$ be a category and let $A$ be an object of $C$ . Then we define $C_{A}$ to be the category whose objects and morphisms are as follows:

$Obj (C_{A}) =$ the set of all morphisms from any object in $C$ tothe object $A$ . Thus, $f \in Obj (C_{A}) ⟺ f \in {Hom}_{C} (Z, A) for some object Z \in Obj (C)$ .
For any two objects $f_{1} : Z_{1} \to A$ and $f_{2} : Z_{2} \to A$ in $C_{A}$ , ${Hom}_{C_{A}} (f_{1}, f_{2})$ contains any morphism $σ \in {Hom}_{C} (Z_{1}, Z_{2})$ such that $f_{2} σ = f_{1}$ .

The back side has an diagram of a slice category. Can you remember what it looks like? There is also info on co-slice categories. Can you remember the definition?

Here is diagram representing two objects $f_{1}$ and $f_{2}$ of $C_{A}$ .

Here is a diagram of the same two objects including a morphism in ${Hom}_{C_{A}} (f_{1}, f_{2})$ . The morphism $σ$ of $C_{A}$ is a morphims of $C$ such that $f_{1} = f_{2} σ$ .

Aluffi describes the morphisms of $C_{A}$ as being commutative diagrams in the ambient cantegory, where ambient category refers to $C$ .

It is worthwhile checking that the category $C_{A}$ as described is indeed a valid category, according to the definition of a category.

Coslice category

Coslice categories are similar to slice categories, and we make use of this similarity to give a compressed definition below.

Let $C$ be a category and $A$ be an object of $C$ . A coslice category $C_{A}$ is one whose objects which are morphisms in $C$ from the object $A$ to another object in $C$ . The morphisms of $C_{A}$ are the morphims of $C$ such that they form commutative diagrams with two objects in $C_{A}$ .

Here is a diagram representing two objects, $f_{1}$ and $f_{2}$ , and a morphism $σ$ of a coslice category where $A$ is an object of the ambient category.

Example

Slice category example

Let $C$ be the category determined by a relation $R$ on the set of integers:

$Obj (C) = Z$ ;
for any objects $a, b \in Z$ , ${Hom}_{C} (a, b) = {(a, b)} if a R b, else {}$ (the entry in the graph representation of the $\leq$ relation at row $a$ and column $b$ ).

Then let $C_{3}$ be the slice category where the objects are the morphisms of $C$ whose target is $3$ . $(0, 3), (- 4, 3), (2, 3),$ and $(3, 3)$ are examples of objects of $C_{3}$ . For objects $(n, 3)$ and $(m, 3)$ in $C_{A}$ (where $n, m \leq 3$ ), there is one morphism from $(n, 3)$ to $(m, 3)$ as long as $n \leq m$ ; that morphism will be $(n, m)$ , which also exists in category $C$ from $n \to m$ .

Aluffi mentions that the category $C_{3}$ can be "harmlessly identified" with the 'subcategory' of integers $\leq 3$ ."

Coslice category example

Define a category, denoted as ${Set}^{*}$ , to be a coslice category where the ambient category is the category of sets-functions, $Set$ , and the target, $A \in Obj (Set)$ , is some singleton set, denoted as ${*}$ .

An object of ${Set}^{*}$ is a function $f : {*} \to S$ , for some set $S$ . The datum of such an object is both the choice of a non-empty set $S$ and a single element within $S$ what is mapped to the single element of the singleton set. Thus, objects of ${Set}^{*}$ can be denoted as set-element pairs, $(S, s)$ , where $S$ is a set and $s \in S$ . Such pairs are called pointed sets.
A morphism of ${Set}^{*}$ , between two objects $(S, s)$ and $(T, t)$ is any function $σ : S \to T$ with the requirement that $σ$ sends $s$ to $t$ . So, if both $S$ and $T$ and not singletons, then, then there will be multiple morphisms between $(S, s)$ and $(T, t)$ .

01.10.2020