typical contexts
The Freyd cover $\overline{\mathcal{T}}$ of a topos $\mathcal{T}$, or more generally of a category with a terminal object, is a way to turn $\mathcal{T}$, viewed as an ‘abstract’ category, into a ‘concrete’ category of structured sets by sewing it together with Set along the global section functor.
As $\overline{\mathcal{T}}$ comes equipped with two well-behaved projection functors the resulting category has many good logical properties making the construction an important tool in type theory and theoretical computer science.
The Freyd cover is sometimes known as the Sierpinski cone or scone, because in topos theory it behaves similarly to the cone on a space, but with the interval $[0,1]$ replaced by the Sierpinski space.
Let $\mathcal{T}$ be a category with a terminal object and $\Gamma = \mathcal{T} (1, -)$ the global section functor . The Freyd cover of $\mathcal{T}$ is the category $\overline{\mathcal{T}}$ whose objects are triples $(X, \xi, U)$ where:
A morphism from $(X,\xi, U)$ to $(Y,\eta , V)$ is a pair of morphisms $(\varphi :X\to Y, t:U\to V)$ with $\varphi\in Set$ and $t\in \mathcal{T}$ such that $\eta\varphi=\Gamma(t)\xi$.
The construction is a special case of Artin gluing: i.e. $\overline{\mathcal{T}}$ is the comma category $Set \downarrow \Gamma$ with $\Gamma = \mathcal{T} (1, -)$.
One of the first applications of the Freyd cover was to deduce facts about the initial topos? (initial with respect to logical morphisms — also called the free topos). They were originally proved by syntactic means; the conceptual proofs of the lemma and theorem below are due to Freyd.
For any category $C$ with a terminal object $\mathbf{1}$, the terminal object of the Freyd cover $\widehat{C}$ is small-projective, i.e., the representable $\Gamma = \widehat{C}(1, -) \colon \widehat{C} \to Set$ preserves any colimits that exist.
To check that $\Gamma^{op} \colon \widehat{C}^{op} \to Set^{op}$ preserves limits, it suffices to check that the composite
preserves limits, because the contravariant power set functor $P = 2^-$ is monadic. But it is easily checked that this composite is the contravariant representable given by $(2, \mathbf{1}, 2 \to \Gamma(\mathbf{1}))$.
The terminal object in the initial topos $\mathcal{T}$ is connected and projective in the sense that $\Gamma = \hom(1, -) \colon \mathcal{T} \to Set$ preserves finite colimits.
We divide the argument into three segments:
The hom-functor preserves finite limits, so by general properties of Artin gluing, the Freyd cover $\widehat{\mathcal{T}}$ is also a topos. Observe that $\mathcal{T}$ is equivalent to the slice $\widehat{\mathcal{T}}/M$ where $M$ is the object $(\emptyset, \mathbf{1}, \emptyset \to \Gamma(\mathbf{1}))$. Since pulling back to a slice is a logical functor, we have a logical functor
Since $\mathcal{T}$ is initial, $\pi$ is a retraction for the unique logical functor $i \colon \mathcal{T} \to \widehat{\mathcal{T}}$.
We have maps $\mathcal{T}(1, -) \to \widehat{\mathcal{T}}(i 1, i-) \cong \widehat{\mathcal{T}}(1, i-)$ (the isomorphism comes from $i 1 \cong 1$, which is clear since $i$ is logical), and $\widehat{\mathcal{T}}(1, i-) \to \mathcal{T}(\pi 1, \pi i-) \cong \mathcal{T}(1, -)$ since $\pi$ is logical and retracts $i$. Their composite must be the identity on $\mathcal{T}(1, -)$, because there is only one such endomorphism, using the Yoneda lemma and terminality of $1$.
Finally, since $\mathcal{T}(1, -)$ is a retract of a functor $\widehat{\mathcal{T}}(1, i-)$ that preserves finite colimits (by the lemma, and the fact that the logical functor $i$ preserves finite colimits), it must also preserve finite colimits.
This is important because it implies that the internal logic of the free topos (which is exactly “intuitionistic higher-order logic”) satisfies the following properties:
The disjunction property: if “P or Q” is provable in the empty context, then either P is so provable, or Q is so provable. (Note that this clearly fails in the presence of excluded middle.)
The existence property: if “there exists an $x\in A$ such that $P(x)$” is provable in the empty context, then there exists a global element $x\colon 1\to A$ such that $P(x)$ is provable in the empty context. (Again, this is clearly a constructivity property.)
The negation property: False is not provable in the empty context.
All numerals in the free topos are “standard”, i.e., the global sections functor $\Gamma = \hom(1, -): \mathcal{T} \to Set$ preserves the natural numbers object $N$ (because $N$ can be characterized in terms of finite colimits and $1$, by a theorem of Freyd).
The Freyd cover of a topos is a local topos, and in fact freely so. Every local topos is a retract of a Freyd cover.
See (Johnstone, lemma C3.6.4).
The Freyd cover $\overline{\mathcal{T}}$ is fibered over $\mathcal{T}$ as it arises equivalently by change of base of the codomain fibration of Set along the global section functor $\Gamma$.
See (Jacobs, p.57).
Some of the above material is taken from
The following two n-café blog posts provide a nice discussion of the scone construction from a geometric and a logical perspective:
Mike Shulman, Discreteness, Concreteness, Fibrations, and Scones , November 2011. (link)
Mike Shulman, Scones, Logical Relations, and Parametricity , April 2013. (link)
You can find more on Artin gluing in this important (and nice) paper:
plus
See also section C3.6 of
Peter Johnstone, Sketches of an Elephant vol. 2 . Oxford UP 2002.
Bart Jacobs, Categorical Logic and Type Theory , Elsevier Amsterdam 1999.
The argument given above for properties of the free topos is an amplification of
For more on the free topos and the first appearance in print of Freyd’s observation see