A bicategory is a notion of weak 2-category that arises as a category weakly enriched in categories. That is, instead of having hom sets, between any two objects a bicategory has hom categories such that the enriched category laws hold up to invertible 2-cell rather than strictly.

A bicategory $\mathrm{𝒦︀}$ consists of

1. A type of objects ${\mathrm{𝒦︀}}_0$, or 0-cells
2. For all $𝑥,𝑦:{\mathrm{𝒦︀}}_0$, a category ${\mathrm{𝒦︀}}_1(𝑥,𝑦)$. We may elide the subscript and simply write this as $\mathrm{𝒦︀}(𝑥,𝑦)$. Refer to the objects of $\mathrm{𝒦︀}(𝑥,𝑦)$ as 1-cells between $𝑥$ and $𝑦$, and we may write $𝑓:\mathrm{𝒦︀}(𝑥,𝑦)$ as $𝑓:𝑥\to 𝑦$ or $𝑥\stackrel{𝑓}{\to }𝑦$. For $𝑓,𝑔:\mathrm{𝒦︀}(𝑥,𝑦)$, refer to the morphisms in $\mathrm{𝒦︀}(𝑥,𝑦)$ between $𝑓$ and $𝑔$ as 2-cells and write the morphism $𝛼:(\mathrm{𝒦︀}(𝑥,𝑦))(𝑓,𝑔)$ as $𝛼:𝑓\Rightarrow 𝑔$ or $𝑓\stackrel{𝛼}{\Rightarrow }𝑔$
3. For each $𝑥:{\mathrm{𝒦︀}}_0$, an identity 1-cell $1_{𝑥}:\mathrm{𝒦︀}(𝑥,𝑥)$
4. For all $𝑥,𝑦,𝑧:{\mathrm{𝒦︀}}_0$, a composition functor ${\mathrm{𝒦︀}}_{\star }^{𝑥,𝑦,𝑧}:\mathrm{𝒦︀}(𝑥,𝑦)\times \mathrm{𝒦︀}(𝑦,𝑧)\to \mathrm{𝒦︀}(𝑥,𝑧)$. For 1-cells $𝑓:𝑥\to 𝑦$ and $𝑔:𝑦\to 𝑧$, write their composite as $𝑓\star 𝑔:𝑥\to 𝑧$

5. For all $𝑤,𝑥,𝑦,𝑧:{\mathrm{𝒦︀}}_0$, a natural isomorphism, the associator $𝛼$ between the two composite functors $\mathrm{𝒦︀}(𝑤,𝑥)\times \mathrm{𝒦︀}(𝑥,𝑦)\times \mathrm{𝒦︀}(𝑦,𝑧)\to \mathrm{𝒦︀}(𝑤,𝑧)$ that compose the leftmost, respectively rightmost, pair first:

   $${\mathrm{𝒦︀}}_{\star }^{𝑤,𝑦,𝑧}\circ ({\mathrm{𝒦︀}}_{\star }^{𝑤,𝑥,𝑦}\times \text{id})\Rightarrow {\mathrm{𝒦︀}}_{\star }^{𝑤,𝑥,𝑧}\circ (\text{id}\times {\mathrm{𝒦︀}}_{\star }^{𝑥,𝑦,𝑧})$$

   Its component at 1-cells $𝑓,𝑔,ℎ$ is the invertible 2-cell

   $${𝛼}_{𝑓,𝑔,ℎ}:(𝑓\star 𝑔)\star ℎ\Rightarrow 𝑓\star (𝑔\star ℎ)$$

6. For all $𝑥,𝑦:{\mathrm{𝒦︀}}_0$, natural isomorphisms, the left unitor $𝜆$ and right unitor $𝜌$, each between an endofunctor of $\mathrm{𝒦︀}(𝑥,𝑦)$ and the identity functor:

   $${\mathrm{𝒦︀}}_{\star }^{𝑥,𝑥,𝑦}\circ ⟨1_{𝑥},\text{id}⟩\Rightarrow \text{id}{\mathrm{𝒦︀}}_{\star }^{𝑥,𝑦,𝑦}\circ ⟨\text{id},1_{𝑦}⟩\Rightarrow \text{id}$$

   where $1_{𝑥}$ and $1_{𝑦}$ in the pairings $⟨-,-⟩$ denote the constant functors at the identity 1-cells. The components at a 1-cell $𝑓:\mathrm{𝒦︀}(𝑥,𝑦)$ are the invertible 2-cells

   $${𝜆}_{𝑓}:1_{𝑥}\star 𝑓\Rightarrow 𝑓{𝜌}_{𝑓}:𝑓\star 1_{𝑦}\Rightarrow 𝑓$$

7. such that for all $𝑓:\mathrm{𝒦︀}(𝑥,𝑦)$ and $𝑔:\mathrm{𝒦︀}(𝑦,𝑧)$ the triangle below commutes in $\mathrm{𝒦︀}(𝑥,𝑧)$:

8. and such that for all composable 1-cells $𝑓,𝑔,ℎ,𝑘$ the pentagon below commutes:

Fix a bicategory $\mathrm{𝒦︀}$, with composition $\star$, identity 1-cells $1_{𝑥}$, associator $𝛼$, and unitors $𝜆,𝜌$. A monad in $\mathrm{𝒦︀}$ internalises the usual notion of monad: it is an endo-1-cell carrying a multiplication and a unit that satisfy the monoid laws up to the coherence cells of the bicategory.

A monad in $\mathrm{𝒦︀}$ consists of

1. a 0-cell $𝑥$, the object the monad acts on;
2. an endo-1-cell $𝑡:\mathrm{𝒦︀}(𝑥,𝑥)$;
3. a multiplication 2-cell $𝜇:𝑡\star 𝑡\Rightarrow 𝑡$;
4. a unit 2-cell $𝜂:1_{𝑥}\Rightarrow 𝑡$;

5. such that $𝜇$ is associative: the following diagram of 2-cells commutes in $\mathrm{𝒦︀}(𝑥,𝑥)$, where the top map is the associator that rebrackets the threefold composite:

6. and such that $𝜇$ and $𝜂$ satisfy the unit laws: the following two diagrams commute in $\mathrm{𝒦︀}(𝑥,𝑥)$, where the hypotenuses are the left and right unitors:

Taking $\mathrm{𝒦︀}$ to be the bicategory of categories, functors, and natural transformations recovers an ordinary monad on a category: $𝑡$ is the endofunctor, $𝜇$ the multiplication, and $𝜂$ the unit, with the coherence cells all identities.

The bicategory of categories $\text{𝖢𝖠𝖳}$ has

• as 0-cells, categories (at a fixed pair of universe levels, for objects and for morphisms);
• as hom-category $\text{𝖢𝖠𝖳}(\mathrm{𝒞︀},\mathrm{𝒟︀})$, the functor category $[\mathrm{𝒞︀},\mathrm{𝒟︀}]$, so 1-cells are functors and 2-cells are natural transformations;
• as identity 1-cell, the identity functor;

• as composition, $𝐹\star 𝐺=𝐺\circ 𝐹$ on functors. On natural transformations $𝛽:𝐹\Rightarrow 𝐹\prime$ and $𝛾:𝐺\Rightarrow 𝐺\prime$ the horizontal composite is given directly by its components

  $${(𝛽\star 𝛾)}_{𝑐}=𝐺({𝛽}_{𝑐})\star {𝛾}_{𝐹\prime 𝑐}.$$

Every component of the left unitor, the right unitor and the associator is an identity morphism, and their inverses are identities too. So the only content of the triangle and pentagon is that composites of identities are identities.

$\text{𝖢𝖠𝖳}$ is still a bicategory and not a strict 2-category: $\text{Id}\circ 𝐹$ and $𝐹$ agree on objects and on morphisms, but in the formalization they are not the same functor definitionally. The structure cells are there to name that agreement.

A monad in $\text{𝖢𝖠𝖳}$ is an ordinary monad on a category, and a prestack is a pseudofunctor into $\text{𝖢𝖠𝖳}$.

A lax functor $𝐹:\mathrm{ℬ︀}\to \mathrm{𝒞︀}$ between bicategories consists of

1. a map on 0-cells, $𝑥\mapsto 𝐹𝑥$;
2. for all $𝑥,𝑦$, a functor ${𝐹}_{𝑥,𝑦}:\mathrm{ℬ︀}(𝑥,𝑦)\to \mathrm{𝒞︀}(𝐹𝑥,𝐹𝑦)$, acting on 1-cells and 2-cells;
3. a unit comparison, natural 2-cells ${𝐹}_{𝑥}^0:1_{𝐹𝑥}\Rightarrow 𝐹(1_{𝑥})$;
4. a composition comparison, 2-cells ${𝐹}_{𝑓,𝑔}^2:𝐹𝑓\star 𝐹𝑔\Rightarrow 𝐹(𝑓\star 𝑔)$ natural in $𝑓$ and $𝑔$;

5. such that three coherence laws hold, one for each structure cell of $\mathrm{ℬ︀}$:

   • left unit: $({𝐹}^0▷𝐹𝑓)\star {𝐹}_{1,𝑓}^2\star 𝐹({𝜆}_{𝑓})={𝜆}_{𝐹𝑓}$;
   • right unit: $(𝐹𝑓◁{𝐹}^0)\star {𝐹}_{𝑓,1}^2\star 𝐹({𝜌}_{𝑓})={𝜌}_{𝐹𝑓}$;
   • associativity: $({𝐹}_{𝑓,𝑔}^2▷𝐹ℎ)\star {𝐹}_{𝑓\star 𝑔,ℎ}^2\star 𝐹({𝛼}_{𝑓,𝑔,ℎ})={𝛼}_{𝐹𝑓,𝐹𝑔,𝐹ℎ}\star (𝐹𝑓◁{𝐹}_{𝑔,ℎ}^2)\star {𝐹}_{𝑓,𝑔\star ℎ}^2$.

Here $\star$ between 2-cells is vertical composition, and $𝜃▷ℎ$ and $ℎ◁𝜃$ are whiskerings.

Lax functors compose: ${(𝐺\circ 𝐹)}^0={𝐺}^0\star 𝐺({𝐹}^0)$ and ${(𝐺\circ 𝐹)}_{𝑓,𝑔}^2={𝐺}_{𝐹𝑓,𝐹𝑔}^2\star 𝐺({𝐹}_{𝑓,𝑔}^2)$. The coherence laws of the composite follow from those of $𝐹$ and $𝐺$ and naturality of ${𝐺}^2$, without using the triangle or pentagon of any of the bicategories involved.

A lax functor whose comparison cells are invertible is a pseudofunctor.

Let $𝐹,𝐺:\mathrm{ℬ︀}\to \mathrm{𝒞︀}$ be lax functors. A lax natural transformation $𝜎:𝐹\Rightarrow 𝐺$ consists of

1. for each 0-cell $𝑥$, a 1-cell ${𝜎}_{𝑥}:𝐹𝑥\to 𝐺𝑥$;

2. for each 1-cell $𝑓:𝑥\to 𝑦$, a 2-cell filling the naturality square,

   $${𝜎}_{𝑓}:𝐹𝑓\star {𝜎}_{𝑦}\Rightarrow {𝜎}_{𝑥}\star 𝐺𝑓;$$
3. such that ${𝜎}_{𝑓}$ is natural in $𝑓$: for a 2-cell $𝜃:𝑓\Rightarrow 𝑔$, $(𝐹𝜃▷{𝜎}_{𝑦})\star {𝜎}_{𝑔}={𝜎}_{𝑓}\star ({𝜎}_{𝑥}◁𝐺𝜃)$;
4. and such that $𝜎$ respects the comparison cells of $𝐹$ and $𝐺$: one law relating ${𝜎}_{1_{𝑥}}$ to ${𝐹}^0$, ${𝐺}^0$ and the unitors, and one relating ${𝜎}_{𝑓\star 𝑔}$ to ${𝜎}_{𝑓}$, ${𝜎}_{𝑔}$, ${𝐹}^2$, ${𝐺}^2$ and the associators.

A lax natural transformation is pseudonatural when every ${𝜎}_{𝑓}$ is invertible. As with pseudofunctors, this is a property, so the pseudonatural transformations $𝐹\Rightarrow 𝐺$ are a full subcategory of the category of lax transformations and modifications.

Between prestacks the 1-cells are taken to be pseudonatural. The reason is biuniversality: a transformation whose components ${𝜎}_{𝑥}$ are all equivalences of categories is an equivalence of prestacks only if its naturality cells are invertible, and a biuniversal element should be exactly a representation of a prestack up to such an equivalence.

Every category $\mathrm{𝒞︀}$ is a bicategory $\text{𝖫𝖣}(\mathrm{𝒞︀})$ with only identity 2-cells. Its 0-cells are the objects of $\mathrm{𝒞︀}$, and the hom-category $\text{𝖫𝖣}(\mathrm{𝒞︀})(𝑥,𝑦)$ is the discrete category on the set $\mathrm{𝒞︀}(𝑥,𝑦)$: a 2-cell $𝑓\Rightarrow 𝑔$ is a proof that $𝑓=𝑔$. Composition and identities are those of $\mathrm{𝒞︀}$; the unitors and associator are the unit and associativity laws of $\mathrm{𝒞︀}$. Since the homs of $\mathrm{𝒞︀}$ are sets, any two parallel 2-cells are equal, so the triangle and pentagon hold trivially.

A functor $𝐹:\mathrm{𝒞︀}\to \mathrm{𝒟︀}$ gives a pseudofunctor $\text{𝖫𝖣}(𝐹):\text{𝖫𝖣}(\mathrm{𝒞︀})\to \text{𝖫𝖣}(\mathrm{𝒟︀})$, whose comparison 2-cells are the functor laws of $𝐹$.

The locally discrete bicategory is how ordinary indexed categories enter bicategorical language: a prestack on $\text{𝖫𝖣}(\mathrm{𝒞︀})$ is a pseudofunctor ${\mathrm{𝒞︀}}^{\text{op}}\to \text{𝖢𝖠𝖳}$, and its Grothendieck construction is a displayed category over $\mathrm{𝒞︀}$.

The opposite ${\mathrm{ℬ︀}}^{\text{op}}$ of a bicategory $\mathrm{ℬ︀}$ has the same 0-cells and reverses the 1-cells but not the 2-cells:

Composition swaps its arguments, $𝑓{\star }^{\text{op}}𝑔=𝑔\star 𝑓$. The left unitor of ${\mathrm{ℬ︀}}^{\text{op}}$ is the right unitor of $\mathrm{ℬ︀}$ and vice versa, and the associator of ${\mathrm{ℬ︀}}^{\text{op}}$ is the inverse of the associator of $\mathrm{ℬ︀}$, with its arguments reversed.

Since the 2-cells keep their direction, a lax functor $𝐹:\mathrm{ℬ︀}\to \mathrm{𝒞︀}$ induces a lax (not oplax) functor ${\mathrm{ℬ︀}}^{\text{op}}\to {\mathrm{𝒞︀}}^{\text{op}}$ with the same action on cells. Reversing the 2-cells instead gives the bicategory ${\mathrm{ℬ︀}}^{\text{co}}$, whose hom-categories are the opposites ${(\mathrm{ℬ︀}(𝑥,𝑦))}^{\text{op}}$.

Duality saves work: a coherence lemma about $\mathrm{ℬ︀}$ can often be obtained by instantiating a companion lemma at ${\mathrm{ℬ︀}}^{\text{op}}$, which swaps left and right.

A pseudofunctor $𝐹:\mathrm{ℬ︀}\to \mathrm{𝒞︀}$ is a lax functor whose unit and composition comparisons

are invertible 2-cells. So $𝐹$ preserves identities and composition up to coherent isomorphism.

Being pseudo is a property of a lax functor: invertibility of a 2-cell is a proposition, since inverses are unique. The data of a pseudofunctor is exactly the data of a lax functor, and everything proved about lax functors applies to pseudofunctors unchanged. Pseudofunctors are closed under composition and identities, as lax functors are, because invertible 2-cells are closed under composition and under the action of a functor on hom-categories.

The main examples here are prestacks, pseudofunctors ${\mathrm{ℬ︀}}^{\text{op}}\to \text{𝖢𝖠𝖳}$. When $\mathrm{ℬ︀}$ is locally discrete on a category $\mathrm{𝒞︀}$, these are the pseudofunctors ${\mathrm{𝒞︀}}^{\text{op}}\to \text{𝖢𝖠𝖳}$ of fibred category theory.

The total bicategory of a displayed bicategory over $\mathrm{𝒦︀}$ packages the base and the displayed data together, one dimension up from the total category of a displayed category.

• Its 0-cells are pairs of a 0-cell of $\mathrm{𝒦︀}$ and a displayed 0-cell over it.
• Its hom-category from to is the total category of the displayed hom-category . So a 1-cell is a pair and a 2-cell is a pair .
• Identities, composition, unitors and associator are pairs of the base structure and the displayed structure over it, and the triangle and pentagon hold because they hold in the base and, over that, in the displayed bicategory.

Projecting to first components is a pseudofunctor whose unit and composition comparisons are identity 2-cells.