Pseudofunctor

Idea

A pseudofunctor is a weakened version of a functor between 2-categories, where preservation of identities and composition holds only up to specified coherent isomorphism, rather than strictly.

A pseudofunctor $F:\mathcal{C}\to \mathcal{D}$ between 2-categories assigns objects, 1-cells, and 2-cells of $\mathcal{C}$ to those of $\mathcal{D}$, preserving vertical composition strictly but preserving identities and horizontal composition only up to specified coherent invertible 2-cells.

Equivalently, a pseudofunctor is a weak 2-functor.

Definition

Let $\mathcal{C},\mathcal{D}$ be 2-Category. A pseudofunctor $F:\mathcal{C}\Rightarrow \mathcal{D}$ consists of:

• On objects: an assignment $X\mapsto FX$.
• On hom-categories: for each $X,Y$, a functor $F_{X,Y}:\mathcal{C}[X,Y]\Rightarrow \mathcal{D}[FX,FY].$ We write $Ff$ for the image of a 1-cell $f\in \mathcal{C}[X,Y]$, and $F\alpha$ for the image of a 2-cell $\alpha$.
• Invertible 2-cells (coherence isomorphisms):
  • Unitors ${\iota }_X:1_{FX}\Rightarrow F(1_X)$, natural in $X$.
  • Compositors ${\mu }_{g,f}:Fg\circ Ff\Rightarrow F(g\circ f)$, natural in $f,g$.

These data satisfy the standard coherence axioms:

• Associativity (pentagon): for composable $f,g,h$, the two 2-cells $((Fh\circ Fg)\circ Ff)\stackrel{\alpha }{\Rightarrow }(Fh\circ (Fg\circ Ff))\stackrel{{\mu }_{h,g}\circ \mathrm{id}}{\Rightarrow }F(h\circ g)\circ Ff\stackrel{{\mu }_{h\circ g,f}}{\Rightarrow }F(h\circ g\circ f)$ and $((Fh\circ Fg)\circ Ff)\stackrel{{\mu }_{h,g}\circ \mathrm{id}}{\Rightarrow }F(h\circ g)\circ Ff\stackrel{{\mu }_{h\circ g,f}}{\Rightarrow }F(h\circ g\circ f)$ agree (with the associator $\alpha$ in $\mathcal{D}$ inserted where appropriate in a bicategory).
• Unitality (triangles): for $f\in \mathcal{C}[X,Y]$, $(Ff\circ 1_{FX})\stackrel{\mathrm{id}\circ {\iota }_X}{\Rightarrow }(Ff\circ F{1}_X)\stackrel{{\mu }_{f,1_X}}{\Rightarrow }F(f\circ 1_X)\stackrel{F({\rho }_f)}{\Rightarrow }Ff,$ $(1_{FY}\circ Ff)\stackrel{{\iota }_Y\circ \mathrm{id}}{\Rightarrow }(F{1}_Y\circ Ff)\stackrel{{\mu }_{1_Y,f}}{\Rightarrow }F(1_Y\circ f)\stackrel{F({\lambda }_f)}{\Rightarrow }Ff,$ where ${\lambda }_f,{\rho }_f$ are the unitors in $\mathcal{C}$ (trivial if $\mathcal{C}$ is strict).

Variance determines the direction of $\mu$:

• Covariant $F:\mathcal{B}\Rightarrow \mathbf{Cat}$: ${\mu }_{g,f}:Fg\circ Ff\Rightarrow F(g\circ f)$.
• Contravariant $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$: ${\mu }_{g,f}:Pf\circ Pg\Rightarrow P(g\circ f)$, so $P(g\circ f)\cong Pf\circ Pg$.

If all coherence 2-cells are identities, $F$ is a (strict) 2-functor.

Coherence data and naturality

• The unitors ${\iota }_X$ are natural in $X$ and compatible with whiskering.
• The compositors ${\mu }_{g,f}$ are natural in $f,g$ and compatible with whiskering on both sides.
• The pentagon and triangle axioms imply all higher coherence equalities, ensuring compositions defined via $\mu$ are well behaved up to canonical isomorphism.

Variants

• Lax functor: ${\iota }_X$ and ${\mu }_{g,f}$ need not be invertible.
• Oplax functor: ${\iota }_X:F(1_X)\Rightarrow 1_{FX}$ and ${\mu }_{g,f}:F(g\circ f)\Rightarrow Fg\circ Ff$ (arrows reversed).
• Pseudofunctor between bicategories: same data, with associators/unitors of the bicategories appearing in the axioms.

Examples

• Indexed categories and reindexing (contravariant):
  • Given a Grothendieck Fibration $p:\mathcal{E}\Rightarrow \mathcal{B}$ with a cleavage, the fibres ${\mathcal{E}}_b$ and reindexing functors $u^{\ast }:{\mathcal{E}}_b\Rightarrow {\mathcal{E}}_{b^{\prime }}$ assemble into a pseudofunctor $(-{)}^{\ast }:{\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}.$ Composition $v^{\ast }\circ u^{\ast }\cong (u\circ v{)}^{\ast }$ gives the compositor.
• Indexed op-categories and pushforward (covariant):
  • For an Opfibration $p:\mathcal{E}\Rightarrow \mathcal{B}$ with op-cleavage, pushforwards $u_{!}:{\mathcal{E}}_b\Rightarrow {\mathcal{E}}_{b^{\prime }}$ assemble into a pseudofunctor $(-{)}_{!}:\mathcal{B}\Rightarrow \mathbf{Cat}.$
• Simple/Grothendieck construction:
  • For $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$, the projection $\pi :\int P\Rightarrow \mathcal{B}$ is a Simple Fibration (hence a Grothendieck fibration).
  • For $Q:\mathcal{B}\Rightarrow \mathbf{Cat}$, $\int Q\Rightarrow \mathcal{B}$ is a split opfibration.
• Discrete case:
  • presheaves $F:{\mathcal{B}}^{op}\Rightarrow \mathbf{Set}$ correspond to Discrete Fibrations via the category of elements $\int F\Rightarrow \mathcal{B}$.
• 2-categorical generalisation:
  • Street Fibrations over a 2-category $\mathcal{B}$ correspond (up to biequivalence) to pseudofunctors ${\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$ via the 2-categorical Grothendieck construction.

Notation

• We write $F:\mathcal{C}\Rightarrow \mathcal{D}$ for a pseudofunctor between 2-categories (using $\Rightarrow$ for functors between categories).
• For contravariant $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$, the compositor is often written ${\mu }_{g,f}:Pf\circ Pg\stackrel{\cong }{\Rightarrow }P(g\circ f),{\iota }_b:1\stackrel{\cong }{\Rightarrow }P(1_b).$

Remarks

• A pseudofunctor between strict 2-categories need not be strict. Strictness of the functorial action need not be forced by strictness of the source/target 2-categories.

Related Concepts

Grothendieck Fibration Opfibration Cloven Fibration Split Fibration Simple Fibration Street Fibration Cartesian Fibration Cocartesian Fibration Category 2-Category Bicategory

References

• https://ncatlab.org/nlab/show/pseudofunctor
• https://ncatlab.org/nlab/show/Grothendieck+construction