Questions: 2-Categories and Weak Functors

Question 1

In the 2-category Cat, what play the roles of 0-cells, 1-cells, and 2-cells respectively?

• A. Categories, functors, and natural transformations
• B. Objects, morphisms, and functors between categories
• C. Sets, functions, and natural transformations between them
• D. Functors, natural transformations, and modifications between natural transformations

Answer: (A)

Cat is the canonical example of a strict 2-category. Its 0-cells are categories (the 'objects' at the top level), its 1-cells are functors between categories (the 'morphisms'), and its 2-cells are natural transformations between functors. Composition of 1-cells is functor composition (strictly associative), and composition of 2-cells is vertical composition of natural transformations. The 2-categorical framework unifies these three layers — which were previously described separately — into a single coherent structure where categorical concepts like equivalence and adjunction become statements about 1-cells and 2-cells.

Question 2

A mathematician observes that the functor assigning to each ring its category of modules only preserves composition 'up to natural isomorphism' rather than strictly. She concludes this construction is defective and should be replaced by one that is strictly associative. This conclusion is:

• A. Wrong — most naturally occurring constructions only preserve structure up to coherent isomorphism, making weak functors the appropriate and more general framework; the failure of strict preservation is a feature of mathematical reality, not a defect
• B. Correct — strict 2-functors are always preferable because they are easier to work with and just as expressive in all practical situations
• C. Wrong, but only because the ring-modules construction can always be strictified with sufficient bookkeeping
• D. Correct — if a functor fails to preserve composition strictly, it technically violates the definition of a functor and cannot be used as a 2-functor

Answer: (A)

In mathematics, most naturally occurring functorial constructions (tensor product of modules, pullback of sheaves, etc.) preserve composition only up to coherent isomorphism. The 2-categorical framework accommodates this through weak (pseudo) functors, which specify explicit comparison 2-cells φ_{g,f}: F(g∘f) → F(g)∘F(f) satisfying coherence conditions. This is not a compromise — it is the correct description of the mathematical reality. While strictification theorems sometimes allow replacing a weak functor with a strict equivalent, this often destroys naturality or simplicity. Weak functors are the standard, not the exception.

Question 3

A strict 2-functor F between 2-categories satisfies F(g ∘ f) = F(g) ∘ F(f) exactly on the nose, whereas a weak 2-functor only guarantees an invertible 2-cell F(g ∘ f) ≅ F(g) ∘ F(f).

• T. True
• F. False

Answer: True

This is precisely the distinction between strict and weak 2-functors. A strict 2-functor preserves all structure at the level of equality: F of a composite equals the composite of F's. A weak (pseudo) functor replaces equality with specified invertible 2-cells φ_{g,f}: F(g∘f) → F(g)∘F(f), together with coherence conditions (a pentagon for triple composites and unit conditions for identities) ensuring the comparison 2-cells are compatible with associativity. The coherence conditions are essential: without them, 'preservation up to isomorphism' would be ill-defined because different re-bracketings of the same composite could give different results.

Question 4

In a 2-category, most 2-cells (morphisms between morphisms) should be invertible for the structure to be well-defined.

• T. True
• F. False

Answer: False

2-cells in a 2-category need not be invertible. In Cat, the 2-cells are natural transformations, and most natural transformations are not natural isomorphisms — they do not have inverses as 2-cells. The 2-category axioms only require that 2-cells compose (horizontally and vertically) in ways consistent with associativity and identity laws; they impose no invertibility requirement. A 2-groupoid is the special case where all 1-cells and 2-cells are invertible, but this is a restricted substructure. Weak functors specify invertible comparison 2-cells, but this is a property of the functor, not a requirement on the ambient 2-category.

Question 5

What is the significance of coherence conditions in weak functors? Why can't we simply assert that F(g ∘ f) is isomorphic to F(g) ∘ F(f) without specifying additional requirements?

Model answer: A weak functor specifies, for each composable pair (f, g), an invertible 2-cell φ_{g,f}: F(g∘f) → F(g)∘F(f). But when three 1-cells h, g, f are composable, there are two distinct ways to re-bracket the triple composite using the comparison 2-cells: one can apply φ_{h,g∘f} then (id_F(h) ∗ φ_{g,f}), or apply φ_{h∘g,f} then (φ_{h,g} ∗ id_F(f)), and these must agree. The coherence condition — a pentagon diagram analogous to the pentagon in monoidal categories — ensures that all re-bracketings of any composite give the same result. Without this, 'preservation up to isomorphism' would be ambiguous: the functor's action on composites would depend on the order of re-bracketing, destroying functoriality.

Coherence is what makes the isomorphisms canonical rather than arbitrary. It is the 2-categorical analogue of the associativity and unit axioms in ordinary category theory, lifted one dimension. The same pattern repeats in higher category theory: at each new dimension, equality is replaced by a specified equivalence one dimension up, with coherence conditions ensuring consistency. Understanding this pattern at the 2-categorical level gives the conceptual vocabulary for ∞-categories and homotopy type theory.