Let $P: \mathcal{C}\to\mathcal{S}$ a fibration ($\mathcal{S}$ with finite limits).

In "Sketches of an Elephant I", pag. 272 P. Johnstone define $P$ a locally small if:

given any two objects $X, Y\in \mathcal{C}$ (let $A=P(X),\ B=P(Y)$) there exist a arrow $(a, b): I\to A\times B$ and a morphisms $f: a^\ast X\to b^\ast Y$ in the fibre $\mathcal{C}(I)$ such that given any $(c, d): J\to A\times B$ and any $g: c^\ast X\to d^\ast Y $ in $\mathcal{C}(J)$ there exists a unique $u: J\to I$ such that $a\circ u=c,\ b\circ u=d$ and $u^\ast(f)\ \dot{=}\ g$ (where " $\dot{=}$ " means up to canonical isomorphisms).

In LNM 661 "Indexed Categories and its Applications" p. 40, the authors post the J. Benabou definition: $P$ is a locally small if:

for any $I\in\mathcal{S}$ and every $X, Y\in \mathcal{C}(I)$ there exist a morfisms ${}^Ih_{A, B}: {}^IH_{A, B} \to I$ such that for any $\alpha: J\to I$ there is a bijection between the morphisms:

$\alpha \to {}^Ih_{A, B}$ in $\mathcal{S}\downarrow I$

and the morphisms $\alpha^\ast(X)\to \alpha^\ast(Y)$ in $\mathcal{C}(I)$

I ask: how are related (if they are) these two definitions?


Johnston define "locally small" as the (what he define) comprehension scheme for the inclusion $2\to \underline{2}$

($2$ is the discrte category $\{0, 1}$ and $\underline{2}$ is $0\to 1$ plus identities).

THis means that the composition funtor $Rect(\underline{2},\ \mathcal{C} )\to Rect(2,\ \mathcal{C} )$ has a right adjoint, where $Rect(\mathcal{D},\ \mathcal{C} )$ is the category with objects the diagrams $d: \mathcal{D}\to \mathcal{C}$ with vertical edges (i.e. mapped to identities by $P$), and with morphisms the transformations with all components cartesians. (articulating this, get the definition given at the beginning).

THe BEnabou definition is equivalent to the more strict assert:

the composition funtor $Rect_P(\underline{2},\ \mathcal{C} )\to Rect_P(2,\ \mathcal{C} )$ has a right adjoint, where $Rect_P(\mathcal{D},\ \mathcal{C} )\subset Rect(\mathcal{D},\ \mathcal{C} )$ is given by diagram $d: \mathcal{D}\to \mathcal{C}$ with $P\circ d$ constant (i.e. maps all on some object $A$ and its identity $1_A$) and with morphisms the transformations with all components cartesians and mapped by $P$ on the same morphism.

THen the Johnstone definition imply the BEnabou one: by restriction of adjunction, observing that $2$ is just the objects of $\underline{2}$ then the condiction on transformations components is preserved.

From the CHuck answere, the reverse i true too:

LEt $X, Y\in \mathcal{C}$ and let $A=P(X),\ B=P(Y)$, considering $\pi_A: A\times B\to A,\ \pi_B : A\times B\to B$, put $I:=A\times B$ and considering $\pi_A^\ast(X),\ \pi_B^\ast(Y)$ on the $I$-fibre, appling the BEnabou condiction follow the Johnstone one.

Then the initial request has the follow generalization:

given a functor $F: \mathcal{D'}\to \mathcal{D} $

Is true that:

If the natural funtor

$Rect(\mathcal{D},\ \mathcal{C} )\to Rect(\mathcal{D'},\ \mathcal{C} )$ has a right adjoint


$Rect_P(\mathcal{D},\ \mathcal{C} )\to Rect_P(\mathcal{D'},\ \mathcal{C} )$ has a right adjoint ?

(from above, I seem the if $F$ is surjective on objets then the proposition is true)

Is true the reverse?, with such conditions on $F$ the reverse can be true?

  • $\begingroup$ I'm fairly sure they're equivalent. See Theorem 10.1 in Streicher's notes, or Theorem B2.2.2 in the Elephant. $\endgroup$
    – Zhen Lin
    Aug 19, 2012 at 13:19
  • $\begingroup$ THe definition in Streicher notes is equivalent to above BEanbou definition (see also "Categoriacal Logic and Type theory" B. JAcobs, Lemma 9.5.4, pag. 561) $\endgroup$ Aug 19, 2012 at 14:36
  • $\begingroup$ I may be misunderstanding your notation here, but your definition of $Rect_P$ seems to me to be the same as $Rect$ since all diagrams in $Rect$ already lie in only one fiber and have only vertical edges and hence applying $P$ to them will 'crush' them to the same object and its identity. $\endgroup$
    – Chuck
    Aug 19, 2012 at 19:18
  • $\begingroup$ on a non cennected cathegory as $2$, $Rect$ can send $0$ and $1$ on differents fibre, or also if send them on the some fibre, the 2 components of a transformations can have different priections on the base category. Anyway I seems that the above generalization is true if $\mathcal{D'}\subset \mathcal{D}$ and $\mathcal{D'}$ has finite connect components. $\endgroup$ Aug 19, 2012 at 22:04

1 Answer 1


The definitions are indeed equivalent. The idea of 'local smallness' is to get for any $X$,$Y$ in $\mathcal{C}^I$ an object of your indexing category to represent, as it were, all (vertical) morphisms between $X$ and $Y$. Both definitions describe this fact, although Johnstone's is, I guess, slightly more 'general' than it needs to be in that it applies the above property to any $X$ and $Y$ in $\mathcal{C}$ (and not to $X$, $Y$ in the same fibre), but that's OK by Theorem 10.1 in Streicher since $\mathcal{S}$ has finite limits. The equivalence of the definitions can also be proved as exercises 8.8.9 and 8.8.10 in Volume 2 of Borceaux.

To see that they are equivalent let for simplicity $X$ and $Y$ lie on the same fibre $I$ (WLOG bearing in mind what I said above.) Then Johnstone's definition says that there exists an arrow $\alpha \colon J \rightarrow I$ and a morphism $f \colon \alpha^*X \rightarrow \alpha^*Y$ such that for any $\beta \colon K \rightarrow I$ and any morphism $g \colon \beta^*X \rightarrow \beta^*Y$ there exists a unique $u: K \rightarrow J$ such that

$$u^{*}(f) = g$$

and $\alpha \circ u = \beta$. But now if you write $J$ as $H_{X,Y}$ and $\alpha$ as $h_{X,Y}$ you'll see that the last sentence says exactly that there is a bijection between morphisms $f \colon \beta^*X \rightarrow \beta^*Y$ and morphisms $u$ such that $h_{X,Y} \circ u = \beta$, i.e. between morphisms from $\beta$ to $h_{X,Y}$ in $\mathcal{S}/I$. And this is exactly the second definition.


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