This post was triggered by John Armstrong’s comment on my last post.
We need to distinguish two ideas: representations of a mathematical concept and the total concept. (I will say more about terminology later.)
Example: We can construct the quotient of the kernel of a group homomorphism by taking its cosets and defining a multiplication on them. We can construct the image of the homomorphism by take the set of values of the homomorphism and using the multiplication induced by the codomain group. The quotient group and the image are the same mathematical structure in the sense that anything useful you can say about one is true of the other. For example, it may be useful to know the cardinality of the quotient (image) but it is not useful to know what its elements are.
But hold on, as the Australians say, if we knew that the codomain was an Abelian group then we would know that the quotient group was abelian because the elements of the image form a subgroup of the codomain. (But the Australians I know wouldn’t say that.)
Now that kind of thinking is based on the idea that the elements of the image are “really” elements of the codomain whereas elements of the quotients are “really” subsets of the domain. That is outmoded thinking. The image and the quotient are the same in all important aspects because they are naturally isomorphic. We should think of the quotient as just as much as subgroup of the codomain as the image is. John Baez (I think) would say that to ask whether the subgroup embedding is the identity on elements or not is an evil question.
Let’s step back and look at what is going on here. The definition of the quotient group is a construction using cosets. The definition of the image is a construction using values of the homomorphism. Those are two different specific representations of the same concept.
But what is the concept, as distinct from its representations? Intuitively, it is
- All the constructions made possible by the definition of the concept.
- All the statements that are true about the concept.
(That is not precise.)
The total concept is like the clone plus the equational theory of a specific type of algebra in the sense of universal algebra. The clone is all the operations you can construct knowing the given signature and equations and the equational theory is the set of all equations that follow from them. That is one way of describing it. Another is the monad in Set that gives the type of algebra — the operations are the arrows and the equations are the commutative diagrams.
Note: The preceding description of the monad is not quite right. Also the whole discussion omits mention of the fact that we are in the world (doctrine) of universal algebra. In the world of first order logic, for example, we need to refer to the classifying topos of the category of algebras of that type (or to its first order theory).
We need better terminology for all this. I am not going to propose better terminology, so this is a shaggy dog story.
Math ed people talk about a particular concept image of a concept as well as the total schema of the concept.
In categorical logic, we talk about the sketch or presentation of the concept vs. the theory. The theory is a category (of the kind appropriate to the doctrine) that contains all the possible constructions and commutative diagrams that follow from the presentation.
In this post I have used “total concept” to refer to the schema or theory. I have referred the particular things as “representations” (for example construct the image of a homomorphism by cosets or by values of the homomorphism).
“Representation” does not have the same connotations as “presentation”. Indeed a presentation of a group and a representation of a group are mathematically two different things. But I suspect they are two different aspects of the same idea.
All this needs to be untangled. Maybe we should come up with two completely arbitrary words, like “dostak” and “dosh”.