ct.category theory - Freyd cover of a category.

I don't know anything about it myself, but here are some other phrases you might try looking up.

The Freyd cover of a category is sometimes known as the Sierpinski cone, or "scone". It's also a special case of Artin gluing. Given a category $mathcal{T}$ and a functor $F: mathcal{T} to mathbf{Set}$, the Artin gluing of $F$ is the comma category $mathbf{Set}downarrow F$ whose objects are triples $(X, xi, U)$ where:

• $X$ is a set


• $T$ is an object of $mathcal{T}$


• $xi$ is a function $X to F(U)$.

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

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.

(Incidentally, my Oxford English Dictionary tells me that the correct spelling is 'gluing', but some people, such as these authors, use 'glueing'. I'm sure Peter Johnstone has a reason.)