<p>I found the reference for CPO-like categories.
Scott's paper <em>Continuous Lattices</em> in the book <em>Toposes, Algebraic Geometry and Logic</em>. It is explained in the comments right after corollary 4.3.
A more general theorem can be found in Smyth's and Plotkin's paper <em>Category-Theoretic Solution of Recursive Domain Equations</em>. It is lemma 2.</p>

<p>However, again, the functors are not arbitrary. One needs some sort of a continuity assumption. </p>

CC BY-SA 4.0
2022-11-17T03:56:22.753Z
Lemma 2 there is the same as Adamek's construction of an initial algebra using omega-colimits, it doesn't say anything about final coalgebras.
Max New
CC BY-SA 3.0
2014-03-03T10:07:20.730Z
2014-03-03T10:07:20.730Z
sonat
<p>I read Freyd's paper "Algebraically Complete Categories" in the famous Como90 and I have two questions about the notion of algebraic compactness he defined in that paper. (If you are not familiar with the definition, here it is: A category is called algebraically compact if every endofunctor has an initial algebra and a final co-algebra which are canonically isomorphic.)</p>

<ol>
<li><p>What are some examples of algebraically compact categories? Freyd mentions an example but strictly speaking the condition in the definition holds only for certain endofunctors of interest. From reading other papers (such as "Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire") I guess that category of cpo's, omega-cpo's or categories enriched over (omega-) cpo's are algebraically compact. What is the standard reference for this fact?</p></li>
<li><p>Freyd says that the definition is motivated by the "principal of versality" and, being a nonnative speaker of English, I am confused. First of all, I think it should be principle, not principal. Also what is versality? Does he mean versatility? Is this a game on words like (uni)versality?</p></li>
</ol>

CC BY-SA 3.0
2014-02-17T13:33:47.907Z
Not being an expert in "Algebraically Complete Categories," I don't want to make this an answer, but being a native English speaker... on your #2, "principal" appears to be a complete typo, especially since he misuses the word again, but in a different grammatical context, in the following sentence as well. He should have used "principle." On the other hand, "versality" -- from the word "versal" -- is an (archaic) shortening of "universality"/"universal". Now, I'm not one to argue with an author NAMING things, but it //appears// he meant to say "Principle of Universality"
Daniel Apon
CC BY-SA 3.0
2014-02-17T13:39:04.503Z
Let me amend the above: "versality" may have a _formal_ definition distinct from "universality" in your context; please check for this. :) For example, see the appendix of http://arxiv.org/pdf/1109.6093v4.pdf
Daniel Apon
CC BY-SA 3.0
2014-03-16T04:44:45.857Z
I agree that "versal" is not the same as "universal". For example there is the notion of [versal deformation](http://en.wikipedia.org/wiki/Milnor_number#Versal_Deformations) in singularity theory, roughly it means that all possible deformations are included, but perhaps not uniquely, i. e. they may occur several times.
მამუკა ჯიბლაძე
CC BY-SA 3.0
2014-03-16T05:02:01.667Z
I think it is especially important to distinguish these in computer science. E. g. for most enumerable sets, every possible enumeration hits infinitely many elements of the set infinitely many times. One-to-one ("*uni*versal") enumerations are rare.
მამუკა ჯიბლაძე
<reference-request><type-theory><ct.category-theory>
Algebraically Compact Categories
sonat
# Algebraically Compact Categories

(Asked by: username_0 on 2014-02-13)

I read Freyd's paper "Algebraically Complete Categories" in the famous Como90 and I have two questions about the notion of algebraic compactness he defined in that paper. (If you are not familiar with the definition, here it is: A category is called algebraically compact if every endofunctor has an initial algebra and a final co-algebra which are canonically isomorphic.)

1. What are some examples of algebraically compact categories? Freyd mentions an example but strictly speaking the condition in the definition holds only for certain endofunctors of interest. From reading other papers (such as "Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire") I guess that category of cpo's, omega-cpo's or categories enriched over (omega-) cpo's are algebraically compact. What is the standard reference for this fact?
2. Freyd says that the definition is motivated by the "principal of versality" and, being a nonnative speaker of English, I am confused. First of all, I think it should be principle, not principal. Also what is versality? Does he mean versatility? Is this a game on words like (uni)versality?

**Tags:** reference-request, type-theory, ct.category-theory

> **Comments:**
> **username_1**: Not being an expert in "Algebraically Complete Categories," I don't want to make this an answer, but being a native English speaker... on your #2, "principal" appears to be a complete typo, especially since he misuses the word again, but in a different grammatical context, in the following sentence as well. He should have used "principle." On the other hand, "versality" -- from the word "versal" -- is an (archaic) shortening of "universality"/"universal". Now, I'm not one to argue with an author NAMING things, but it //appears// he meant to say "Principle of Universality"  
> **username_1**: Let me amend the above: "versality" may have a _formal_ definition distinct from "universality" in your context; please check for this. :) For example, see the appendix of http://arxiv.org/pdf/1109.6093v4.pdf  
> **username_2**: I agree that "versal" is not the same as "universal". For example there is the notion of [versal deformation](http://en.wikipedia.org/wiki/Milnor_number#Versal_Deformations) in singularity theory, roughly it means that all possible deformations are included, but perhaps not uniquely, i. e. they may occur several times.  
> **username_2**: I think it is especially important to distinguish these in computer science. E. g. for most enumerable sets, every possible enumeration hits infinitely many elements of the set infinitely many times. One-to-one ("*uni*versal") enumerations are rare.  

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## 1 answers

### Answer by username_0 on 2014-03-03. Score: 3

I found the reference for CPO-like categories.
Scott's paper *Continuous Lattices* in the book *Toposes, Algebraic Geometry and Logic*. It is explained in the comments right after corollary 4.3.
A more general theorem can be found in Smyth's and Plotkin's paper *Category-Theoretic Solution of Recursive Domain Equations*. It is lemma 2.

However, again, the functors are not arbitrary. One needs some sort of a continuity assumption.

> **Comments:**
> **username_3**: Lemma 2 there is the same as Adamek's construction of an initial algebra using omega-colimits, it doesn't say anything about final coalgebras.  

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