I apologize in advance for my naiveté. I'm completely missing the formal background to ask this question, but am very curious about one particular point.
I recently learned that there's the concept of a 2-category, which has objects, morphisms, and morphisms between morphisms. I didn't go through the formal definition, but I immediately got the feeling that this must be a useful concept, because of the following two familiar examples of categories admitting "morphisms between morphisms": in category theory itself, there are categories, functors, and natural transformations between functors. Even more impressive, in topology there are topological spaces, continuous maps, and homotopies between continuous maps. I will call this informally the "2-category of topological spaces".
In homotopy theory one is interested in topological spaces up to the following notion of equivalence: we identify two topological spaces $X$ and $Y$ if there are two continuous maps $f\colon X\to Y$ and $g\colon Y\to X$ such that $f\circ g$ is homotopic to the identity on $Y$ and $g\circ f$ is homotopic to the identity on $X$. In this case we say that $X$ and $Y$ are homotopy equivalent.
This construction can be cast in categorical terms by considering what Wikipedia calls the naive homotopy category: the objects are topological spaces and the morphisms are homotopy classes of continuous maps. This is a well-defined category because of the following property (also taken from Wikipedia) that I am going to call $(*)$:
if $f_1, g_1 \colon X → Y$ are homotopic, and $f_2, g_2 \colon Y → Z$ are homotopic, then their compositions $f_2 \circ f_1$ and $g_2 \circ g_1 \colon X → Z$ are also homotopic.
Now the following holds (by definition of the notion of an isomorphism in a category): two topological spaces are isomorphic in the naive homotopy category if and only if they are homotopy equivalent.
Note that this construction can be considered as a passage from the "2-category of topological spaces" to the naive homotopy category: morphisms in the latter are equivalence classes of morphisms in the former (and the notion of "equivalence of morphisms" is induced by the morphisms between morphisms). So we started with a 2-category and pressed it into a conventional category by identifying "isomorphic" morphisms in the 2-category.
Question 1: Does this construction of a category from a 2-category (by identifying isomorphic morphisms) have a name?
In order for this to work, I guess the following generalization of $(*)$ should hold in every 2-category: if $f_1, g_1 \colon X → Y$ are isomorphic, and $f_2, g_2 \colon Y → Z$ are isomorphic, then their compositions $f_2 \circ f_1$ and $g_2 \circ g_1 \colon X → Z$ are also isomorphic, where $f$ is isomorphic to $g$ if there are "morphisms between morphisms" $\eta\colon f\to g$ and $\lambda\colon g\to f$ with $\eta\circ \lambda$ equals the identity "morphism between morphisms" on $g$ and $\lambda \circ \eta$ equals the identity "morphism between morphisms" on $f$. (If that makes sense.)
Question 2: Does this generalization of $(*)$ hold in every 2-category?
Question 3: Let's call the above construction of a category from a 2-category the homotopy category of the 2-category. If two 2-categories $C$ and $D$ are biequivalent (I read on the nLab that this "is the appropriate notion of equivalence between 2-categories"), does it follow that the homotopy category of $C$ is equivalent to the homotopy category of $D$?
I already did some research regarding Question 1, but the only thing I could find is the notion of a homotopy 2-category. This seems very related: it's about assigning a 2-category to any "(∞,n)-category" (I guess that's a category with even morphisms between morphisms of morphisms and so on). But I can't find anything on making a conventional category from a 2-category.
Best Answer
I believe "the homotopy category of a 2-category" conventionally refers to the category obtained by taking the 2-category, identifying any 1-morphisms that are isomorphic, and discarding the 2-cells. This is justified because one can define "the homotopy theory of a 2-category" as follows: discard all non-invertible 2-morphisms to obtain a groupoid-enriched category, then take nerves of hom-groupoids to obtain a simplicially enriched category (in fact, enriched in Kan complexes). The homotopy category of a 2-category is thus recovered as a special case of the homotopy category of a simplicially enriched category.
Horizontal composition in a 2-category is functorial, in the sense if you have objects $X, Y, Z$ then you have a horizontal composition functor $\textrm{Hom} (Y, Z) \times \textrm{Hom} (X, Y) \to \textrm{Hom} (X, Z)$. In particular, horizontal composition respects isomorphisms of 1-morphisms.
If a 2-functor 2-fully-faithful (i.e. induces equivalences of hom-categories) then the induced functor between homotopy categories is fully faithful. If a 2-functor is 2-essentially surjective on objects then the induced functor between homotopy categories is essentially surjective on objects. Thus, biequivalent 2-categories have equivalent homotopy categories. The converse is about as far from true as you can get – after all, any 1-category is also a 2-category in a trivial way, but a 2-category is usually not biequivalent to its homotopy category. More seriously, the construction of the homotopy category ignores non-invertible 2-morphisms, which means you also can't distinguish between a 2-category and its "underlying" (2, 1)-category.