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Reflexive Subcategories as Fibrations

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Recently when looking at various fibration constructions, I realised that there is a closed connection between an adjunction of a reflexive subcategory and fibrations. In general, if we have a functor $p : E \to B$ that $E$ has and $p$ preserves pullbacks, where it also admits a fully faithful right adjoint $i : B \to E$ that makes the counit naturally isomorphic to the identity functor on $B$, then $p$ is a fibration. The cartesian maps are natural transformation squares of the unit $\eta$ of the adjunction which is also a pullback: for any $f : x \to y$ in $E$, $f$ is cartesian iff $f$ is the pullback of $p(f)$ along $\eta_y$. $p$ is a fibration because we can pullback any morphism $u : I \to p(y)$ in $B$ (considered included in $E$ using $i$) along the unit $\eta_y : y \to p(y)$, and since $p$ preserves pullback we can prove the resulting pullback squares gives us the desired cartesian lifting of $u$.

There are many examples of this type. For example, let $\partial_1 : B^\to \to B$ be the fundamental fibration of a category $B$ which has pullbacks, the right adjoint takes any object $I$ in $B$ to the identity on $I$, making $B$ a reflexive subcategory of $B^\to$. $\partial_1$ preserves connected limits, hence preserves pullbacks. There are some other trivial examples of this kind: if $C$ has a terminal object, then the projection fibration $ \pi_B : B \times C$ has a right adjoint, sending each $I$ in $B$ to $(I,1)$ in $B \times C$, making $B$ a reflexive subcategory. The projection obviously preserves pullbacks. Another perhaps more interesting case is the fibration of polynomial functors $\mathbf{Poly}$ onto $\mathbf{Set}$: the evaluation on the singleton set $ -(1) : \mathbf{Poly} \to \mathbf{Set}$ is a fibration, and it has both a left and right adjoint, both of which are fully faithful. The right adjoint of it sends each set $A$ to the constant functor on $A$, and it makes $\mathbf{Set}$ a reflexive subcategory of $\mathbf{Poly}$. All these fibration structures are supposed to be described by the general case in the previous paragraph.

The case where it might get interesting for topos theory is that it makes every geometric inclusion of sheaf category into the corresponding presheaf category fibrational. For any site $(\mathcal C,J)$, the sheafification $a : [\mathcal{C}^{\operatorname{op}},\mathbf{Set}] \to \mathbf{Sh}(\mathcal C,J)$ is actually a fibration according to our previous discussion, with fibres of a sheaf $\mathcal F$ being the collection of all presheaves whose sheafification is isomorphic to $\mathcal F$, and $\mathcal F$ can be viewed as a canonical colimit of this fibre, with the unit of the sheafification adjunction being the cocone morphisms. By the way, since a quasi-topos is a reflexive subcategory of a presheaf category whose localisation preserves products, from this connection with fibration it might also be interesting to consider a reflexive subcategory of a presheaf category whose localisation preserves pullbacks.

If I'm not taking these examples wrong, I'm wondering whether this connection goes any further and whether or not it is interesting or has more interesting things to say. Due to my relatively limited background, I also do not know whether these observation has been made or remained folklore or not. If there are literatures on these, I'm also happy to know where I can find more of these related things.

Merci beaucoup!

This topic was modified 2 years ago 3 times by Lingyuan Ye

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Hi Lingyuan!

@Riccardo_Zanfa and I have some joint work in progress on similar ideas! Your observation about fibrations with adjoints can be extended a lot further. It's interesting to examine these properties in the context of geometric morphisms more generally too. I'm not sure how much insight it provides into reflective subcategories, though, unless you have some descriptions of extra properties a fibration can have that would allow you to distinguish further classes of such reflections. Once our work is finalized we'll surely write about it here, so look out for it!

The reason your observation isn't so common is that "fibration" typically refers (in 'pure category theory' contexts) to Grothendieck fibrations, which are strict. The reflection into a left exact reflective subcategory is typically a Street fibration (the definition is only verified up to isomorphism). The latter is the more natural to study as a standalone concept, but the former are everywhere because of the correspondence between cloven Grothendieck fibrations and pseudofunctors.