Limit-sketchability

Limit-sketchability

[Michael Barr]

Does anyone know whether every limit closed full subcategory of an 
equational category is the category of models of a limit sketch? 
Assuming that is true and the equational category is finitary, if the 
subcategory is also closed in the equational category under filtered 
colimits, is also the category of models an FL-sketch?

Michael


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Re: Limit-sketchability

[Jiri Adamek]

Dear Michael,

Your first question

> Does anyone know whether every limit closed full subcategory of an
> equational category is the category of models of a limit sketch?

depends on set theory: if Vopenka's Principle (VP) holds, the answer is
affirmative, see Corrollary 6.24 in "Locally Presentable and
Accessible Categories". If VP does not hold, the answer
is negative. Our counterexample 6.25 works the category Rel \Sigma
which is not equational, but which can embedded as a full, limit-closed
subcategory of an equational category (see e.g. Theorem 1.46 in the book).

> Assuming that is true and the equational category is finitary, if the
> subcategory is also closed in the equational category under filtered
> colimits, is also the category of models an FL-sketch?

The answer is affirmative (absolutely): a full subcategory of a locally
finitely presentable category closed under limits and filtered colimits
is locally finitely presentable, see Corollary 2.48 in the above
book.

Best regards
Jiri


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