Simple Fibration

Definition

Given a pseudofunctor $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$, the Grothendieck construction $\pi :\int P\Rightarrow \mathcal{B}$ is a fibration, called the simple fibration associated to $P$.

• Objects of $\int P$ are pairs $(b,x)$ with $b\in \mathcal{B}$ and $x\in P(b)$.
• A morphism $(b^{\prime },x^{\prime })\to (b,x)$ is a pair $(u,\phi )$ where $u\in \mathcal{B}[b^{\prime },b]$ in $\mathcal{B}$ and $\phi \in P(b^{\prime })[x^{\prime },P(u)(x)]$ in $P(b^{\prime })$.
• The projection $\pi (b,x)=b$ and $\pi (u,\phi )=u$.

For $u\in \mathcal{B}[b^{\prime },b]$ and $(b,x)$, the arrow $(u,{\mathrm{id}}_{P(u)(x)}):(b^{\prime },P(u)(x))\to (b,x)$ is cartesian, giving the cartesian lift of $u$ at $(b,x)$. Hence $\pi$ is a Grothendieck Fibration.

If $P$ is strict (a functor), then these chosen lifts are strictly functorial, making $\pi$ a Split Fibration. For a general pseudofunctor, $\pi$ is a Cloven Fibration with cleavage determined by the choices above, and composition/identities hold up to the coherence isomorphisms of $P$.

Reindexing

Writing the fibre over $b$ as $(\int P{)}_b\cong P(b)$, reindexing along $u\in \mathcal{B}[b^{\prime },b]$ is $u^{\ast }\cong P(u):P(b)\Rightarrow P(b^{\prime }).$

• If $P$ is strict: $({\mathrm{id}}_b{)}^{\ast }=\mathrm{Id}$ and $(u\circ v{)}^{\ast }=v^{\ast }\circ u^{\ast }$ on the nose.
• If $P$ is a pseudofunctor: the equalities hold up to the coherence isomorphisms of $P$.

Relation to Grothendieck fibrations

• Equivalence of data: Giving a cloven Grothendieck Fibration over $\mathcal{B}$ is equivalent (up to equivalence over $\mathcal{B}$) to giving a pseudofunctor $P:{\mathcal{B}}^{op}\Rightarrow \mathbf{Cat}$ via the Grothendieck construction.
• Split replacement: Every Grothendieck fibration is equivalent over $\mathcal{B}$ to a simple fibration arising from a strictification of its reindexing.

Examples

• Family Fibration: Take $\mathcal{B}=\mathbf{Set}$ and $P(B)={\mathbf{Set}}^B$ with reindexing by substitution $u^{\ast }:{\mathbf{Set}}^B\Rightarrow {\mathbf{Set}}^{B^{\prime }}$. Then $\int P\Rightarrow \mathbf{Set}$ is the family (set-indexed) fibration.
• Codomain Fibration: For a category $\mathcal{C}$ with pullbacks, $P(B)=\mathcal{C}/B$ and $P(u)=u^{\ast }:\mathcal{C}/B\Rightarrow \mathcal{C}/B^{\prime }$. The Grothendieck construction $\int P\Rightarrow \mathcal{C}$ is (equivalent to) the codomain fibration $\mathrm{cod}:{\mathcal{C}}^{\to }\Rightarrow \mathcal{C}$.
• Presheaf case (discrete): For a presheaf $F:{\mathcal{B}}^{op}\Rightarrow \mathbf{Set}$, viewing $P(b)$ as a discrete category on $F(b)$ yields $\int F\Rightarrow \mathcal{B}$, a split Discrete Fibration as a special case of a simple fibration.

Related Concepts

• Fibration
• Grothendieck Fibration
• Cloven Fibration
• Split Fibration
• Discrete Fibration
• Opfibration
• Family Fibration
• Codomain Fibration

References

jacobs1999-categorical-logic