# Freyd cover of a category.

I’ve couldn’t find any information about the free category built up from that Freyd cover. Where can I find more about the Freyd cover of a category (not a topos!)?

Edit: The definition has been given in Lambek and Scott's "Higher order categorical logic". I think (according to L. Román) it is initial among all categories endowed with products and a weak nno.

Edit: (Added by Tom Leinster) Here's the definition of Freyd cover, taken from Lambek and Scott (22.1). Let $\mathcal{T}$ be a category with terminal object. Its Freyd cover $\hat{\mathcal{T}}$ is the comma category whose objects are the triples $(X, \xi, U)$ where:

• $X$ is a set
• $U$ is an object of $\mathcal{T}$
• $\xi: X \to \mathcal{T}(1, U)$ is a function.

Lambek and Scott emphasize that $\hat{\mathcal{T}}$ has a terminal object and that it comes equipped with a terminal-object-preserving functor $G: \hat{\mathcal{T}} \to \mathcal{T}$. Strictly speaking, the Freyd cover is the pair $(\hat{\mathcal{T}}, G)$, not just the category $\hat{\mathcal{T}}$ itself.

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Could you give a definition of a Freyd Cover? Maybe if you don't know it in the case you're interested in, give the case you do know, and why you think the definition works in your case somehow? –  Charles Siegel Jan 17 '10 at 19:43
Ximo: I thought it would be helpful to merge your comment (giving the reference) into the main question, so I did it. You might want to delete that comment now. Also, if you don't like the way I've edited your question, you can edit it yourself (and undo my changes if you want). –  Tom Leinster Jan 17 '10 at 21:26

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.)

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