somehow this more obvious name never crossed my mind until now.
Anonymous
]]>new temporary name, surely there’s a better name out there.
Anonymous
]]>I don’t understand post #5; the faithfulness of means that you lose nothing by reinterpreting morphisms of as functions .
]]>Clarifying the last sentence in my previous comment: many (but not all) concrete categories (Set being one of them, but also Ab, CRing, Top, Met, Diff, et cetera) have a function evaluation structure as described in the last sentence of the previous comment, which comes from the fact that group homomorphisms/ring homomorphisms/continuous functions/isometries/differentiable functions are functions of sets which could evaluate an element of a domain to return an element in the codomain.
Of course, in other concrete categories (such as Rel), such a structure doesn’t exist, since relations aren’t functions.
]]>To elaborate on the “concrete” part, the sets of morphisms of any category are already defined as a functor , function composition is already defined as a family of functions in , and a concrete category is defined as a category with a functor , so one could append to the concrete category the structure of function evaluation defined as a family of functions in satisfying certain axioms.
]]>Author of this page: I’m not entirely sure what a good name for such a model would be, I don’t really like the current name myself. I chose “concrete” in part because in the categorical semantics of the existence of collections of elements in the model category is basically the same as a faithful functor , which is the definition of a concrete category, but mostly because I am terrible at coming up with a better name for this model. So if you have any other suggestions for names feel free to list them out.
]]>I may not agree that “concrete” is the right word for this.
I added a query box requesting the anonymous page creator to come discuss at the nForum.
]]>Page created, but author did not leave any comments.
Anonymous
]]>