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 $a : \to \mathbf{Sh}(\mathcal C,J)$ 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!

]]>The resulting model looks a lot different a priori from the model of the finite-limit theory. However, I expect that these should be equivalent somehow (say the categories of models are equivalent). One could phrase this in terms of the extension of the doctrine from finite-limit theories (or whatever fragment one wants: equational, regular, coherent, etc) all the way up to full higher-order logic, and that interpreting the simpler fragment in equivalent ways in the larger fragment leads to something equivalent, no matter how it is convoluted by the richer vocabulary of the larger fragment.

I was wondering if there is anything on this in the literature? Or if it is easy to see? I am only working on instinct.

]]>This section is mainly targeted to those who are familiar with the language of category theory and are interested in learning about toposes as a means to significantly enrich and deepen the categorical outlook on mathematics.

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