When An Abelian Category Has Enough Flat Objects?

How does category theory address products of objects?

• For example, how would I express a map from multiple objects to another object, analogous to a function of multiple parameters?

• Answer:

  When talking about a "map with multiple parameters", say from objects a and b the correct thing to do is to look at a map from the product [math]a \times b[/math]. For example, if [math]a = \{1,2\}[/math] and [math]b = \{3,4\}[/math], we'd get the product [math]a \times b = \{(1,3), (1,4), (2,3), (2,4)\}[/math]. But the question here is how to define the product in the abstract category theory sense, without reference to pairs or even to viewing the objects as sets. So, what's the key property that [math]a \times b[/math] needs to have? Motivated by the set view, [math]a \times b[/math] needs to have the property that we can get out an element of [math]a[/math] and an element of [math]b[/math]. So there should be two maps (or morphisms) [math]a \times b \to a[/math] and [math]a \times b \to b[/math]. These are called projection maps, and a mathematician might write them as [math]\pi_a[/math] or [math]\pi_b[/math]. A Haskell programmer might write them as fst or snd. Of course, there are a lot of objects with maps to [math]a[/math] and [math]b[/math], so how do we hone in on the right one? Well, somehow we want to say that the maps [math]\pi_a[/math] and [math]\pi_b[/math] are "independent" in some sense. With sets, we needed that [math](x, y) \in a \times b[/math] for every [math]x \in a[/math] and [math]y \in b[/math]. To phrase that in terms of maps: for any object [math]c[/math] with maps [math]f : c \to a[/math] and [math]g : c \to b[/math], we should be able to find a map [math]h : c \to a \times b[/math] such that [math]\pi_a \circ h = f[/math] and [math]\pi_b \circ h = g[/math]. An example might clear this up... back to the sets example. Suppose [math]c = \{0\}[/math], and that [math]f(0) = x[/math] and [math]g(0) = y[/math]. Then [math]h(0) = (x, y)[/math], [math]\pi_a((x,y)) = x[/math] and [math]\pi_b((x,y)) = y[/math]. So this property is basically asserting that [math](x, y) \in a \times b[/math]. We also want to require that [math]c[/math] be unique; otherwise, [math]a \times b[/math] might have two elements that act like [math](x, y)[/math]. So with that motivation, we can generally define: The product [math]a \times b[/math] is the object with maps [math]\pi_a : a \times b \to a[/math] and [math]\pi_b : a \times b \to b[/math] such that for any  object [math]c[/math] with maps [math]f : c \to a[/math] and [math]g : c \to b[/math], there is a unique map [math]h : c \to a \times b[/math] such that [math]\pi_a \circ h = f[/math] and [math]\pi_b \circ h = g[/math]. If the product exists, it is unique up to isomorphism. This kind of construction is common in category theory; it is know as a http://en.wikipedia.org/wiki/Universal_property.