April 17
Hans Schoutens,
CUNY
How to avoid evil? Abolish equality!
Rather than a misguided MAGA slogan, this is an attempt to align model-theory with category theory without using type-dependent logics. In category theory, the notion of isomorphism between categories is too restrictive-as it is also in model-theory-, and is replaced by that of 'equivalence of categories'-and in model-theory by 'elementary equivalence of structures' (I will provide precise definitions in the talk). Now, there is an obvious language L_{cat} in which we can consider a category as a first-order (two-sorted) structure. Alas, the first-order L_cat-theory of a given category C might not be the same as that of an equivalent category D. Properties violating this principle of equivalence are called 'evil' by the experts.
I will argue in this talk that this evil disappears if we just drop our insistence that equality (=) should be a logical symbol (rather than part of the signature). I will describe a fo logic w/o equality L_{homo} in which elementary categories are elementary equivalent. This logic has enough expressive power to state, for instance, the existence of products or equalizers. In fact, for skeletal categories, their L_{homo}-theory and L_{cat}-theory are the same (after the appropriate expansions).
This yields in particular a non-evil version of the universality of the category MOD(T) of models of a theory T: any small enough category C that is L_{homo}-equivalent with MOD(T) 'consists' of models of T.