Recall the notion of a Grothendieck fibration: a functor $p \colon E \to B$ whose fibres $E_b$ are (contravariantly) functorial in $b \in B$.
This idea may be generalized to work in any suitable 2-category $K$, although if the 2-category is not strict, then one has to generalize instead the non-strict notion of Street fibration. The generalized definition can be given in any of several equivalent ways, in such a way that
When $K$ is Cat, strict fibrations in this sense are precisely Grothendieck fibrations, while non-strict ones are precisely Street fibrations.
Fix a finitely complete (non-strict) 2-category $K$. Recall that for any two morphisms $f: A \to C$ and $g: B \to C$, $f/g$ denotes their comma object. Let $f/_{\cong} g$ be their 2-pullback.
A morphism $p \colon E \to B$ in $K$ is a (non-strict) fibration when the following equivalent conditions hold:
$p_* = K(X,p) \colon K(X,E) \to K(X,B)$ is a Street fibration in $Cat$ for each $X \in K$, and for all $f \colon Y \to X$ in $K$
is a morphism of fibrations.
For every morphism $f: X \to B$, the canonical map $i: f/_{\cong} p \to f/p$ has a right adjoint in the slice 2-category $K / X$.
The canonical map $i \colon p \to B/p$ has a right adjoint in $K / B$.
$p \colon E \to B$ is an algebra for the 2-monad $L$ on $K/B$ given by $L p = B/p$.
The induced F-functor $\Sigma_p : K\swarrow E \to K\swarrow B$ between the oplax slice F-categories has a right semi-lax F-adjoint. (See Johnstone.)
If $K$ is a strict 2-category with finite strict 2-limits, then we say that $p \colon E \to B$ is a strict fibration when the corresponding conditions hold where “Street fibration” is replaced by “Grothendieck fibration”, slice 2-categories are replaced by strict-slice 2-categories, comma objects are replaced by strict comma objects, and 2-pullbacks are replaced by strict 2-pullbacks. In this case the 2-monad $L$ is in fact a strict 2-monad, but we do not require $p$ to be a strict algebra, only a pseudoalgebra; strict algebras for $L$ would instead be split fibrations.
Note also that the first definition makes perfect sense regardless of whether $K$ has any limits, although this is not true of the others.
By way of spelling out the first definition, we may define a 2-cell $\eta \colon e \to e' \colon X \to E$ to be $p$-cartesian if $f^*\eta = \eta f$ is $p_*$-cartesian for every $f \colon Y \to X$. Since this definition already incorporates stability under pullback, we can then say that $p$ is a fibration in $K$ if for every 2-cell $\beta \colon b \to p e$ there is a cartesian $\hat\beta \colon e' \to e$ such that $p e' = b$ and $p \hat\beta = \beta$.
The third definition is perhaps the simplest. Of course, it is implied by the second (take $f = 1$), but the converse is also true by the pasting lemma for comma and pullback squares.
We can show that the first and third definitions are equivalent by using the representability of fibrations and adjunctions, plus the following lemma, whose proof can be found at Street fibration.
A functor $p \colon E \to B$ is a cloven Street fibration if and only if the canonical functor $i \colon B \to E/p$ has a right adjoint $r$ in $Cat / B$.
It follows that a morphism $p \colon E \to B$ in any 2-category $K$ is representably a fibration (i.e. satisfies the first definition) if and only if the adjunction $i \dashv r$ exists in $K/B$ (i.e. it satisfies the third condition).
Now we connect the first three conditions with the fourth. Because $K$ is finitely complete, we may form the tricategory $Span(K)$ of spans in $K$. In particular, $K/B$ is equivalent to the hom-2-category $Span(K)(B,1)$. Now recall that $B/p$ can be expressed as a pullback or span composite
Write $\Phi B = B^{\mathbf{2}}\rightrightarrows B$. The functor $L \colon p \mapsto B/p$ is then given by composition: $L p =\Phi B \circ p$. To show that $p$ is a fibration iff it is an $L$-algebra, it suffices to show
Lemma. $\Phi B$ is a colax-idempotent monad in $Span(K)$ with unit $i \colon B \to B^{\mathbf{2}} = B/B$.
Proof. Write $\mathbf{\Delta}$ for the monoidal 2-category whose underlying 1-category is the augmented simplex category (q.v.) $\Delta_a$. Recall that for $n \geq 1$ each $[n] \in \mathbf{\Delta}$ is given by the $(n-1)$-fold composite of the cospan $[1] \to [2] \leftarrow [1]$ with itself, where we take a 0-fold composite to denote the identity $[1] \to [1] \leftarrow [1]$. Recall also that the monoidal structure $[n] \oplus [m]$ on $\mathbf{\Delta}$ is generated by composing the cospans $[n],[m] \colon [1] ⇸ [1]$ (together with the fact that $[0],[1]$ are the initial and terminal objects).
Thus for each $n \geq 1$, there is a span $B^{[n]} \colon B ⇸ B$, which is canonically equivalent to the $(n-1)$-fold composite $B^{[2]} \circ B^{[2]} \circ \cdots \circ B^{[2]} \colon B ⇸ B$, because cotensor preserves finite limits. For the same reason, $B^- \colon Cat^{op}_{fp} \to K$ almost restricts to a monoidal 2-functor $\mathbf{\Delta}^{op} \to Span(K)(B,B)$, except that $[0]$ does not yield a span from $B$ to $B$. Precomposing $B^-$ with the functor $\mathbf{\Delta}^{co} \to \mathbf{\Delta}^{op}$ that sends $[n]$ to $[n+1]$, while $\delta_i \mapsto \sigma_i$ and $\sigma_i \mapsto \delta_{i+1}$ does yield a monoidal 2-functor $\mathbf{\Delta}^{co} \to Span(K)(B,B)$, which by the universal property of $\mathbf{\Delta}$ corresponds to a unique colax-idempotent monad in $Span(K)(B,B)$.
In detail, the monoid object $[0] \to [1] \leftarrow [2]$ in $\mathbf{\Delta}$ is sent to the monad $\Phi B = B^{[2]}$ in $Span(K)$, with structure maps $i = (\sigma^1_0)^* \colon B \to \Phi B$ and $c = (\delta^2_1)^* \colon \Phi B \circ \Phi B \to \Phi B$. Moreover, because of the adjunction $\delta^2_1 \dashv \sigma^2_0 = \sigma^1_0 \oplus [1]$ in $\mathbf{\Delta}$, we have $i \circ \Phi B \dashv c$ in $Span(K)(B,B)$ with identity unit.
It follows that $L = \Phi B \circ -$ is a monad, and is colax-idempotent because, for a span $H \colon B ⇸ A$, we have $\eta_H = i \circ H$, $\mu_H = c \circ H$, and
with identity unit. It is clear, too, that the morphism $i \colon p \to B/p$ as above is given by $\eta_{p^\circ} = i \circ p^\circ$, where $p^\circ$ denotes the obvious span $B ⇸ 1$. Thus, by the definition of a colax-idempotent monad, a morphism $p \colon E \to B$ carries the structure of an $L$-algebra if and only if the unit $i = \eta_{p^\circ} \colon p \to B/p$ has a right adjoint.
For a strict 2-category $K$, let $Fib_K$ be the 2-category of strict fibrations in $K$, morphisms of fibrations, and 2-cells between them. Then the codomain functor $cod : Fib_K \to K$ is a strict 2-fibration. (This is arguably an instance of the microcosm principle.)
Similarly, for any bicategory $K$, the bicategory $Fib_K$ of weak fibrations in $K$, morphisms of fibrations, and 2-cells admits a weak 2-fibration $cod : Fib_K \to K$.
It is easy to show that a composite of fibrations is a fibration. Moreover, if $Fib(X)= Fib_K(X)$ denotes the 2-category of fibrations over $X\in K$, then we have:
A morphism in $Fib(X)$ is a fibration in the 2-category $Fib(X)$ iff its underlying morphism in $K$ is a fibration.
This is a standard result, at least in the case $K=Cat$, and is apparently due to Benabou. References include (Bénabou 1985), (Hermida 1999) and (Jacobs 1999, Chapter 9). Therefore, for any fibration $A\to X$ in $K$ we have $Fib_K(A) \simeq Fib_{Fib_K(X)}(A\to X)$, and similarly for opfibrations. This is a fibrational 2-categorical analogue of the standard equivalence $K/A \simeq (K/X)/(A\to X)$ for ordinary slice categories.
Ross StreetFibrations in bicategories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21 no. 2 (1980), p. 111–160 (numdam).
Mark WeberYoneda structure from 2-toposes, Applied Categorical Structures 15 (2007) pp259-323. doi:10.1007/s10485-007-9079-2 (author’s page)
Peter Johnstone, Fibrations and partial products in a 2-category, Applied Categorical Structures 1 (1993) pp141-179 doi:10.1007/BF00880041
Jean BénabouFibered categories and the foundations of naive category theory, Journal of Symbolic Logic, 50(1) (1985) pp10-37. doi:10.2307/2273784, (JSTOR)
Bart Jacobs, Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics, 141. North-Holland Publishing Co., Amsterdam, 1999. xviii+760 pp. ISBN: 0-444-50170-3 (pdf) (contents)
Claudio Hermida, Some properties of Fib as a fibred 2-category, Journal of Pure and Applied Algebra 134 Issue 1 (1999) pp83-109 doi:10.1016/S0022-4049(97)00129-1, (Core)
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