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 $U : C \to Set$ means that you lose nothing by reinterpreting morphisms $A \to B$ of $C$ as functions $U(A) \to U(B)$.

]]>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 $\epsilon_{A,B}:Mor(A,B) \times U(A) \to U(B)$ 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 $Mor:C \times C \to Set$, function composition $\circ_{A,B,C}:Mor(B,C) \times Mor(A, B) \to Mor(A,C)$ is already defined as a family of functions in $Set$, and a concrete category is defined as a category with a functor $U:C \to Set$, so one could append to the concrete category the structure of function evaluation defined as a family of functions $\epsilon_{A,B}:Mor(A, B) \times U(A) \to U(B)$ in $Set$ 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 $C$ is basically the same as a faithful functor $U:C \to Set$, 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

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