如何基于有限表现群与有限范畴抽象等价关系？

Awesome question! This equivalence relation you're describing is a core concept tying group theory and category theory together, and its abstraction boils down to congruence relations—compatible equivalence relations that let us form quotient structures. Let's break this down step by step:

In finite presented groups, the equivalence classes you mentioned (like $abc=d$ and $a(aa^{-1})bc$ being equivalent) are formalized using group congruences:

• Start with a free group F(S), where S is your set of generators. This group contains all possible "words" made by multiplying generators and their inverses.
• Your defining equations (like $abc=d$) are pairs of words declaring two elements should be equivalent. These pairs generate the smallest congruence relation $\sim$ on F(S):
  • $\sim$ is a standard equivalence relation (reflexive, symmetric, transitive).
  • It's compatible with group multiplication: if $x \sim y$, then $zx \sim zy$ and $xz \sim yz$ for any $z \in F(S)$.
• The finite presented group is exactly the quotient group F(S)/~, where each congruence class is a set of words equivalent under your defining relations. The example $a(aa^{-1})bc$ is equivalent to $abc$ because $aa^{-1}$ is the identity element, trivially congruent to the empty word.

When modeling generators as arrows in a finite category, equivalent arrow chains correspond to category congruences, which let us build quotient categories:

• Start with a free category $\mathcal{F}(G)$, where G is a directed graph (vertices are your category's objects, edges are your generator arrows). The arrows in this free category are all valid compositions of generator arrows—exactly the "arrow chains" you mentioned.
• Your defining equations (like "chain $f_1 \circ f_2$ is equivalent to chain $g$") are pairs of parallel arrows (arrows with the same source and target object). These pairs generate the smallest category congruence $\sim$:
  • For every pair of objects $X,Y$, $\sim$ is an equivalence relation on $\text{Hom}(X,Y)$ (the set of arrows from $X$ to $Y$).
  • It's compatible with category composition: if $f \sim f'$ (from $X$ to $Y$) and $g \sim g'$ (from $Y$ to $Z$), then $g \circ f \sim g' \circ f'$.
• The quotient category $\mathcal{F}(G)/\sim$ has congruence classes as its arrows. When we say two arrow chains form a commutative diagram, we're exactly saying those chains are in the same congruence class—they represent the same arrow in the quotient category.

At the root of both contexts, this equivalence relation is a congruence for an algebraic structure:

• Groups are single-object categories, and category congruences generalize group congruences to multiple objects.
• The key idea is that congruences preserve the structure's operations (group multiplication or category composition) across equivalence classes. This lets the quotient set inherit the same algebraic structure—so the quotient of a free group is a group, and the quotient of a free category is a category.

For a concrete example: if you have a group relation $ab=ba$, this translates to a category congruence where the composite arrow $a \circ b$ is equivalent to $b \circ a$. The quotient category will have these composites as a single arrow, which is exactly what we mean when we say $a$ and $b$ commute in a diagram.

内容的提问来源于stack exchange，提问作者Ben Sprott