Answer by Tristan for How are invariants represented in category theory?

2011-02-05T17:13:15Z

That seems like a perfectly reasonable definition. In addition it admits an obvious (and quite useful) description of invariants in terms of higher category theory, where notions of homotopy are more readily generalized. Think again of homology; it's homeomorphism invariant of course, but more importantly it's homotopy invariant. First we describe homotopy in the context of 2 categories:

Let $C$ be a 2-category. We call a 1-morphism $f:a \to b$ a $homotopy$ $equivalence$ if there exists a 1-morphism $g: b \to a$, and invertible 2-morphisms $\alpha : gf \to \textrm{id}_{a}$ and $\beta : fg \to \textrm{id}_{b}$ .

Then we obtain the following definition:

Let $C, D$ be 2-categories, and let $F: C \to D$ be a 2-functor. We say that $F$ is an $invariant$ if, for any homotopy equivalence $f : a \to b$, $F(a) = F(b)$.

This definition more efficiently captures the trait of invariance under homotopy equivalence. If one wishes to (loosely) describe the "singular axiom", or invariance under weak equivalence, the above definition can be reformulated in terms of $\infty$-categories.

User tristan - MathOverflow

Answer by Tristan for How are invariants represented in category theory?