# Is there a concept of "homotopy category of a 2-category"?

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.

• Yes, I believe this is the conventional meaning of "homotopy category of a 2-category". Yes, horizontal composition in a 2-category respects isomorphisms. Yes, biequivalent 2-categories yield equivalent homotopy categories. Apr 10, 2021 at 23:16
• @ZhenLin Thanks so much! Would you mind copy and pasting your comment as an answer? Optimally with a reference to a place in the literature where the "homotopy category of a 2-category" appears, but I also would accept your answer as it is. Apr 11, 2021 at 10:28
• The following additional question came to my mind: how different can two 2-categories with equivalent homotopy categories be? Apr 11, 2021 at 10:30

2. 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.