# Gurski's Definition of a strict functor of tricategories

In Gurskis definition (page 32 of his thesis) of a strict functor $F$ of tricategories he requires that

$F$ maps the adjoint equivalences $a,l,r$ in the source tricategory to the same adjoint equivalences in the target tricategory.

This means for example $F (a_{fgh}) = a_{FfFgFh}$. (Note that these morphisms are indeed parallel, because of the strictness of $F$.)

However it is not required, that the modifications in the definition of a tricateory $(\pi,\lambda,\mu,\rho)$ are preserved as well. For example $F (\pi _{fghk})$ is not required to be equal to $\pi_{FfFgFhFk}$. So my question is:

Why isn't it required that $(\pi,\lambda,\mu,\rho)$ are preserved? Is this because this is already implied by the other requirements? Or are there strict functors which doesn't preserve the modifications $(\pi,\lambda,\mu,\rho)?$

EDIT

Let me draw some parallel to the definition of strict functors between bicategories: ($*$ denotes composition along objects)

There it is only required that the constraint-2-cells $$F(f)*F(g) \to F(f*g)$$ and $$1_{Fa} \to F(1_a)$$ are identities. That $a,l,r$ are preserved is not demanded by definition, but it follows easily from the functor axioms.

Therefore one might hope that in the case of a strict functor of tricategories it follows somewhat analoge that the "highest" constraint data $\pi,\lambda,\mu,\rho$ are preserved, although it is not demanded by definition.

## 1 Answer

The axioms in the definition of a non-strict trihomomorphism (Definition 3.3.1, page 31-32) already require that the modifications $\pi,\lambda,\mu,\rho$ are preserved "up to" the coherences $\pmb{\chi}, \pmb{\iota}, \omega,\gamma,\delta$. The definition of a strict functor (3.3.3, p33) then requires that $\pmb{\chi}$ and $\pmb{\iota}$ are identities while $\omega,\gamma,\delta$ are "as close to identities as possible" (they can't literally be identities because their domain is not equal to their codomain), which therefore implies that $\pi,\lambda,\mu,\rho$ are "preserved as much as possible".

• Please tell me if I am mistaken: The local strictness of $F$ together with the requirement that $\chi$ is an identity ensures that $F$ preserves all kinds of composition "on the nose", i.e. $F(x \blacksquare y) =F(x) \blacksquare F(y)$ for whatever composition $\blacksquare$ stands for. This, together with the requirement that $a$ gets preserved on the nose ensures that $F(\pi_{fghk})$ and $\pi_{FfFgFhFk}$ are parallel morphisms (same source and target) in the target tricategory. Therefore "$\pi$ is preserved as much as possible" would mean "$\pi$ is preserved on the nose"? – Peter Guthmann Apr 20 '18 at 8:22
• @PeterGuthmann sounds possible, but I'm not confident enough to say definitely yes. I suggest asking Nick directly. – Mike Shulman Apr 21 '18 at 2:34
• I followed your advise and Nick Gurski confirmed that strict tricat-functors indeed preserve also the top level coherence-cells strictly. At the moment I don't understand the reason well enough to write a real answer, though. – Peter Guthmann Apr 23 '18 at 18:43