This is a first draft of an article to eventually appear in abstractmath.
Images and metaphors
To explain a math concept, you need to explain how mathematicians think about the concept. This is what in abstractmath I call the images and metaphors carried by the concept. Of course you have to give the precise definition of the concept and basic theorems about it. But without the images and metaphors most students, not to mention mathematicians from a different field, will find it hard to prove much more than some immediate consequences of the definition. Nor will they have much sense of the place of the concept in math and applications.
Teachers will often explain the images and metaphors with handwaving and pictures in a fairly vague way. That is good to start with, but it’s important to get more precise about the images and metaphors. That’s because images and metaphors are often not quite a good fit for the concept — they may suggest things that are false and not suggest things that are true. For example, if a set is a container, why isn’t the element-of relation transitive? (A coin in a coinpurse in your pocket is a coin in your pocket.)
“A metaphor is a useful way to think about something, but it is not the same thing as the same thing.” (I think I stole that from the Economist.) Here, I am going to get precise with the notion that a function is a map. I am acting like a mathematician in “getting precise”, but I am getting precise about a metaphor, not about a mathematical object.
A function is a map
A map (ordinary paper map) of Minnesota has the property that each point on the paper represents a point in the state of Minnesota. This map can be represented as a mathematical function from a subset of a 2-sphere to 
The Minnesota map-as-function is probably continuous and differentiable, and as is well known it can be angle preserving or area preserving but not both.
So you can say there is a point on the paper that represents the location of the statue of Paul Bunyan in Bemidji. There is a set of points that represents the part of the Mississippi River that lies in Minnesota. And so on.
A function has an image. If you think about it you will realize that the image is just a certain portion of the piece of paper. Knowing that a particular point on the paper is in the image of the function is not the information contained in what we call “this map of Minnesota”.
This yields what I consider a basic insight about function-as-map: The map contains the information about the preimage of each point on the paper map. So:
The map in the sense of a “map of Minnesota” is represented by the whole function, not merely by the image.
I think that is the essence of the metaphor that a function is a map. And I don’t think newbies in abstractmath always understand that relationship.
A morphism is a map
The preceding discussion doesn’t really represent how we think of a paper map of Minnesota. We don’t think in terms of points at all. What we see are marks on the map showing where some particular things are. If it is a road map it has marks showing a lot of roads, a lot of towns, and maybe county boundaries. If it is a topographical map it will show level curves showing elevation. So a paper map of a state should be represented by a structure preserving map, a morphism. Road maps preserve some structure, topographical maps preserve other structure.
The things we call “maps” in math are usually morphisms. For example, you could say that every simple closed curve in the plane is an equivalence class of maps from the unit circle to the plane. Here equivalence class meaning forget the parametrization.
The very fact that I have to mention forgetting the parametrization is that the commonest mathematical way to talk about morphisms is as point-to-point maps with certain properties. But we think about a simple closed curve in the plane as just a distorted circle. The point-to-point correspondence doesn’t matter. So this example is really talking about a morphism as a shape-preserving map. Mathematicians introduced points into talking about preserving shapes in the nineteenth century and we are so used to doing that that we think we have to have points for all maps.
Not that points aren’t useful. But I am analyzing the metaphor here, not the technical side of the math.
Groups are functors
People who don’t do category theory think the idea of a mathematical structure as a functor is weird. From the point of view of the preceding discussion, a particular group is a functor from the generic group to some category. (The target category is Set if the group is discrete, Top if it is a topological group, and so on.)
The generic group is a group in a category called its theory or sketch that is just big enough to let it be a group. If the theory is the category with finite products that is just big enough then it is the Lawvere theory of the group. If it is a topos that is just big enough then it is the classifying topos of groups. The theory in this sense is equivalent to some theory in the sense of string-based logic, for example the signature-with-axioms (equational theory) or the first order theory of groups. Johnstone’s Elephant book is the best place to find the translation between these ideas.
A particular group is represented by a finite-limit-preserving functor on the algebraic theory, or by a logical functor on the classifying topos, and so on; constructions which bring with them the right concept of group homomorphisms as well (they will be any natural transformations).
The way we talk about groups mimics the way we talk about maps. We look at the symmetric group on five letters and say its multiplication is noncommutative. “Its multiplication” tells us that when we talk about this group we are talking about the functor, not just the values of the functor on objects. We use the same symbols of juxtaposition for multiplication in any group, “
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