8.2 Coproducts, coequalizers, and pushouts

Coproducts, coequalizers, and pushouts are essential tools in category theory for combining and relating objects. These constructions allow us to create new objects from existing ones, capturing the idea of "gluing" things together in a universal way.

These concepts have wide-ranging applications in mathematics. From constructing free objects to forming quotient structures, they provide a unified framework for understanding various mathematical constructions across different fields.

Coproducts

Coproducts in categories

Coproducts combine two objects $A$ and $B$ in a category $\mathcal{C}$ to form a new object $A \coprod B$ along with morphisms $i_A: A \to A \coprod B$ and $i_B: B \to A \coprod B$ called canonical injections or coprojections For any object $C$ and morphisms $f: A \to C$ and $g: B \to C$, there exists a unique morphism $h: A \coprod B \to C$ satisfying $h \circ i_A = f$ and $h \circ i_B = g$ Examples of coproducts in familiar categories:

• In the category of sets, the coproduct is the disjoint union of sets (set-theoretic union with elements tagged to indicate their origin set)
• In the category of abelian groups, the coproduct is the direct sum of groups (Cartesian product with componentwise addition)

Universal property of coproducts

The universal property states that for any object $C$ and morphisms $f: A \to C$ and $g: B \to C$, there exists a unique morphism $h: A \coprod B \to C$ making the diagram commute This property characterizes the coproduct up to unique isomorphism Coproducts are a special case of colimits, which generalize the concept of "gluing" objects together via a universal property involving a more general diagram

Coequalizers and Pushouts

Coequalizers and quotient objects

Coequalizers "equalize" two parallel morphisms $f, g: A \to B$ in a category $\mathcal{C}$ by forming an object $Q$ with a morphism $q: B \to Q$ such that $q \circ f = q \circ g$ For any object $C$ and morphism $h: B \to C$ satisfying $h \circ f = h \circ g$, there exists a unique morphism $k: Q \to C$ such that $k \circ q = h$ The morphism $q$ is called the canonical projection Examples of coequalizers and quotient objects:

• In the category of sets, the coequalizer is the quotient set $B/\sim$ under the smallest equivalence relation $\sim$ containing $(f(a), g(a))$ for all $a \in A$
• In the category of groups, the coequalizer is the quotient group $H/N$, where $N$ is the normal subgroup generated by $\{f(a)g(a)^{-1} \mid a \in G\}$

Pushouts as generalized constructions

Pushouts generalize coproducts and coequalizers by forming an object $P$ from two morphisms $f: A \to B$ and $g: A \to C$ in a category $\mathcal{C}$ $P$ comes with morphisms $i_B: B \to P$ and $i_C: C \to P$ such that $i_B \circ f = i_C \circ g$ For any object $Q$ and morphisms $j_B: B \to Q$ and $j_C: C \to Q$ satisfying $j_B \circ f = j_C \circ g$, there exists a unique morphism $h: P \to Q$ such that $h \circ i_B = j_B$ and $h \circ i_C = j_C$ Special cases of pushouts:

• When $A$ is an initial object, the pushout is the coproduct $B \coprod C$
• When $B = C$ and $f = g$, the pushout is the coequalizer of $f$ and $g$

Examples of pushouts in familiar categories:

• In the category of sets, the pushout is the quotient of the disjoint union $(B \coprod C)/\sim$, where $\sim$ is the smallest equivalence relation containing $(f(a), g(a))$ for all $a \in A$
• In the category of groups, the pushout is the quotient of the free product $(H * K)/N$, where $N$ is the normal subgroup generated by $\{f(a)g(a)^{-1} \mid a \in G\}$

Applications of categorical colimits

Coproducts construct free objects:

1. The free group on a set $X$ is the coproduct of $X$ copies of the infinite cyclic group $\mathbb{Z}$
2. The free monoid on a set $X$ is the coproduct of $X$ copies of the natural numbers $\mathbb{N}$ under addition
3. The polynomial ring $R[x]$ over a ring $R$ is the coproduct of countably many copies of $R$ Coequalizers construct quotient objects:
4. The quotient group $G/N$ is the coequalizer of the inclusion $i: N \to G$ and the constant morphism $c: N \to G$ sending every element to the identity
5. The quotient ring $R/I$ is the coequalizer of the inclusion $i: I \to R$ and the zero morphism $0: I \to R$
6. The quotient space $X/\sim$ is the coequalizer of the two projections $p_1, p_2: R \to X$ from the equivalence relation $R \subseteq X \times X$
   Pushouts construct amalgamated free products and fiber products:

• The amalgamated free product $G *_H K$ of groups $G$ and $K$ over a common subgroup $H$ is the pushout of the inclusions $i_G: H \to G$ and $i_K: H \to K$
• The fiber product (or pullback) of morphisms $f: A \to C$ and $g: B \to C$ is the dual notion of a pushout, obtained by "reversing the arrows" in the pushout diagram