nLab
pseudofunctor

• functor

• 2-functor / pseudofunctor

• n-functor

• (∞,1)-functor

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Idea

A pseudofunctor is a specific algebraic notion of weak $2$-functor between bicategories (including strict 2-categories), i.e. a functor which preserves composition and identities only up to coherent specified isomorphism.

In general, there is not much reason to say “pseudofunctor” instead of “functor,” since the only relevant type of functor between arbitrary bicategories is weak. However, if the domain and codomain are known to be strict 2-categories (including ordinary $1$-categories), it can be helpful to say “pseudofunctor” or “weak functor” to emphasize that it is not a strict 2-functor. Note that if the codomain is a $1$-category, then there is no difference.

Pseudo or weak functors are also to be distinguished from lax functors and oplax functors, which preserve identities and composition only up to a transformation in one direction or the other, which may be non-invertible.

An older terminology, which should probably be avoided at all costs, uses “homomorphism of bicategories” for a weak functor and “morphism of bicategories” for a lax one.

Definition

Given bicategories $C$ and $D$, a pseudofunctor (or weak $2$-functor, or just functor) $P:C\to D$ consists of:

• for each object $x$ of $C$, an object $P_x$ of $D$;

• for each hom-category $C(x,y)$ in $C$, a functor $P_{x,y}:C(x,y)\to D(P_x,P_y)$;

• for each object $x$ of $C$, an invertible 2-morphism ($2$-cell) $P_{{id}_x}:{id}_{P_x}\Rightarrow P_{x,x}({id}_x)$;

• for each triple $x,y,z$ of $C$-objects, a isomorphism (natural in $f:x\to y$ and $g:y\to z$) $P_{x,y,z}(f,g):P_{x,y}(f);P_{y,z}(g)\Rightarrow P_{x,z}(f;g)$;

• for each hom-category $C(x,y)$,

  $$\begin{array}{ccc}& & {id}_{P_x};P_{x,y}(f)\\ & {}^{P_{{id}_x};{id}_{P_{x,y}(f)}}⇙& & {⇘}^{{\lambda }_{P_{x,y}(f)}}\\ P_{x,x}({id}_x);P_{x,y}(f)& & & & P_{x,y}(f)\\ & {}_{P_{x,x,y}({id}_x,f)}⇘& & {⇗}_{P_{x,y}({\lambda }_f)}\\ & & P_{x,y}({id}_x;f)\\ \end{array}$$\array { & & \id_{P_x} ; P_{x,y}(f) \\ & {}^{P_{\id_x};\id_{P_{x,y}(f)}}\swArrow & & \seArrow^{\lambda_{P_{x,y}(f)}} \\ P_{x,x}(\id_x) ; P_{x,y}(f) & & & & P_{x,y}(f) \\ & {}_{P_{x,x,y}(\id_x,f)}\seArrow & & \neArrow_{P_{x,y}(\lambda_f)} \\ & & P_{x,y}(\id_x;f) \\ }

  and

  $$\begin{array}{ccc}& & P_{x,y}(f);{id}_{P_y}\\ & {}^{{id}_{P_{x,y}(f)};P_{{id}_y}}⇙& & {⇘}^{{\rho }_{P_{x,y}(f)}}\\ P_{x,y}(f);P_{y,y}({id}_y)& & & & P_{x,y}(f)\\ & {}_{P_{x,y,y}(f,{id}_y)}⇘& & {⇗}_{P_{x,y}({\rho }_f)}\\ & & P_{x,y}(f;{id}_y)\\ \end{array}$$\array { & & P_{x,y}(f) ; \id_{P_y} \\ & {}^{\id_{P_{x,y}(f)};P_{\id_y}}\swArrow & & \seArrow^{\rho_{P_{x,y}(f)}} \\ P_{x,y}(f) ; P_{y,y}(\id_y) & & & & P_{x,y}(f) \\ & {}_{P_{x,y,y}(f,\id_y)}\seArrow & & \neArrow_{P_{x,y}(\rho_f)} \\ & & P_{x,y}(f;\id_y) \\ }

  commute; and

• for each quadruple $w,x,y,z$ of $C$-objects,

  $$\begin{array}{ccc}(P_{w,x}(f);P_{x,y}(g));P_{y,z}(h)& \stackrel{{\alpha }_{P_{w,x}(f),P_{x,y}(g),P_{y,z}(h)}}{\Rightarrow }& P_{w,x}(f);(P_{x,y}(g);P_{y,z}(h))\\ P_{w,x,y}(f,g);{id}_{P_{y,z}(h)}\Downarrow & & \Downarrow {id}_{P_{w,x}(f)};P_{x,y,z}(g,h)\\ P_{w,y}(f;g);P_{y,z}(h)& & P_{w,x}(f);P_{x,z}(g;h)\\ P_{w,y,z}(f;g,h)\Downarrow & & \Downarrow P_{w,x,z}(f,g;h)\\ P_{w,z}((f;g);h)& \underset{P_{w,z}({\alpha }_{f,g,h})}{\Rightarrow }& P_{w,z}(f;(g;h))\\ \end{array}$$\array { \big(P_{w,x}(f) ; P_{x,y}(g)\big) ; P_{y,z}(h) & \overset{\alpha_{P_{w,x}(f),P_{x,y}(g),P_{y,z}(h)}}\Rightarrow & P_{w,x}(f) ; \big(P_{x,y}(g) ; P_{y,z}(h)\big) \\ \mathllap{P_{w,x,y}(f,g);\id_{P_{y,z}(h)}}\Downarrow & & \Downarrow\mathrlap{\id_{P_{w,x}(f)};P_{x,y,z}(g,h)} \\ P_{w,y}(f;g) ; P_{y,z}(h) & & P_{w,x}(f) ; P_{x,z}(g;h) \\ \mathllap{P_{w,y,z}(f;g,h)}\Downarrow & & \Downarrow\mathrlap{P_{w,x,z}(f,g;h)} \\ P_{w,z}\big((f;g);h\big) & \underset{P_{w,z}(\alpha_{f,g,h})}\Rightarrow & P_{w,z}\big(f;(g;h)\big) \\ }

  commutes.

Lax functors

If we remove the requirement that $P_{{id}_x}$ and $P_{x,y,z}(f,g)$ be invertible, then we have the definition of lax functor. If we reverse the direction of these as well, then we have an oplax functor.

History

Historically the term ‘pseudofunctor’ was conceived by Grothendieck who weakened, around 1957, the concept of a contravariant functor from a 1-category to Cat, by effectively replacing the $1$-category Cat by the 2-category $\mathrm{Cat}$ and allowing (contravariant) functoriality up to coherent $2$-cells. This was recorded in his Bourbaki seminar on descent via pseudofunctors. Later in SGA1 Grothendieck (with the assistance of Pierre Gabriel) replaced pseudofunctors in the treatment of descent by more invariant fibered categories. Benabou, in his 1967 treatise introducing bicategories, generalized the pseudofunctors of Grothendieck to pseudofunctors between arbitrary bicategories but under the name ‘homomorphism of bicategories’.

Discussion

Tim: in specifying a pseudo functor $F$ you have to decide whether the isomorphism goes from $F(gf)$ to $F(g)F(f)$ or in the other direction. Of course they are equivalent as each will be inverse to the other. You might say that one is lax and pseudo the other op-lax and pseudo. When specifying the Grothendieck construction for such a functor, which is to be preferred?

Both are about equally represented in the literature that I have seen which gets confusing. (In other words, I’m confused!)

Toby: As you suggest, the two versions are equivalent, so in a way it doesn't make a difference. But it might be nice to settle a convention in case we need it.

Tim: I have been using (for the Menagerie) the idea that there are pseudofunctors presented in two equivalent flavours lax pseudofunctor and oplax ones.

Mike: Well, the natural comparison maps that you get in a Grothendieck fibration go in the “lax” direction $F(g)F(f)\to F(gf)$, since they are induced by the universal property of cartesian arrows. In particular, if you have a functor with “weakly cartesian” liftings that don’t compose, then it is a lax functor. Not a very strong argument, but if we just want some convention it might be a reason to pick lax. I think that making too big a deal out of the difference would be misleading, though.