Contents

Idea

The Freyd cover of a category – sometimes known as the Sierpinski cone or “scone” – is a special case of Artin gluing:

given a category $𝒯$ and a functor $F:𝒯\to \mathrm{Set}$, the Artin gluing of $F$ is the comma category $\mathrm{Set}\downarrow F$ whose objects are triples $(X,\xi ,U)$ where:

• $X$ is a set
• $U$ is an object of $𝒯$
• $\xi$ is a function $X\to F(U)$.

So the Freyd cover is the special case $F=𝒯(1,-)$.

References

You can find more on Artin gluing in this important (and nice) paper:

• Aurelio Carboni, Peter Johnstone, Connected limits, familial representability and Artin glueing , Mathematical Structures in Computer Science 5 (1995), 441–459

plus

• Aurelio Carboni, Peter Johnstone, Corrigenda to ‘Connected limits…’ , Mathematical Structures in Computer Science 14 (2004), 185–187.

Some of the above material is taken from

• Tom Leinster, reply at MathOverflow