From what I've been able to find, psuedofunctors are not-quite-functors, in the sense that they preserve the identity morphism and composition of morphisms only up to coherent isomorphism, and not 'on the nose'. But I'm struggling to find an explicit definition of what this means anywhere. The nLab has a large section on coherence theorems, but most of it is far above my head.

I'm assuming that it simply means 'isomorphic in a nice way', but I can't quite pin down what 'nice' means here. Could anybody please explain what it does mean here? Or even give a different definition of a pseudofunctor.

  • $\begingroup$ Are you familiar with 2-categories or bicategories at all, or are you only interested in pseudofunctors to $\mathrm{Cat}$? $\endgroup$ – user54748 Jan 5 '16 at 21:43
  • $\begingroup$ @user54748 I know the basic definitions of 2-categories, but yes, I'm really just interested in pseudofunctors to $\mathsf{Cat}$ $\endgroup$ – Tim Jan 5 '16 at 21:52

The definition of pseudofunctors on nLab is actually quite explicit, but it's for pseudofunctors between bicategories. Borceux (Handbook of Categorical Algebra, volume 1, 7.5) gives the definition for strict categories.

As for what a coherent (iso)morphism is, it's a morphism that satisfies coherence laws. These laws ensure that things that ought to be equal (and trivially would be, if the morphisms were identities) really are. For example in this case you are given isomorphisms $φ : Ff ∘ Fg ≅ F(f ∘ g)$, but to do anything, you'll obviously need isomorphisms $Ff ∘ Fg ∘ Fh ≅ F(f ∘ g ∘ h)$ too, and similarly for any number morphisms. Of course you can get from $Ff ∘ Fg ∘ Fh$ to $F(f ∘ g ∘ h)$ using $φ$, but you can do so in two ways: via $Ff ∘ F(g ∘ h)$ or via $F(f ∘ g) ∘ Fh$. One of the coherence laws for pseudofunctors says that these two ways are the same. That given this, all paths (constructed from $φ$) from $F(f_1) ∘ ... ∘ F(f_n)$ to $F(f_1 ... f_n)$ are the same is then a part of the coherence theorem for pseudofunctors (but these can also be stated in more sophisticated ways).

As a side-note, monoidal categories are a good way to get familiar with coherence and coherence theorems. In a way, anybody who uses product of sets or tensor products already is.

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    $\begingroup$ I have a modest amount of experience with monoidal categories, and was expecting the answer to be something similar to this, but wasn't sure of the details. Thank you for this great answer! $\endgroup$ – Tim Jan 5 '16 at 22:52
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    $\begingroup$ @Tim: In that case, you can just take the definition of a strong monoidal functor, replace objects with 1-morphisms, tensor with composition, and morphisms with 2-morphisms, and you'll get the coherence laws for a pseudofunctor between two bicategories. If you want it strict, just forget about the associators and unitors. $\endgroup$ – user54748 Jan 5 '16 at 23:07

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