Left/Right Issues, Part 1

Any time a mathematical concept comes in a “left-handed” and a “right-handed” flavor, I can almost guarantee that I’m going to have trouble remembering which is which.  I’ll either have to write it somewhere I’ll see it every day until, after a few months, I pick it up via osmosis, or I’ll have to find some way of relating it to the small number of chiral concepts that I actually understand.

Here’s one such way.  We’ll learn to tell the difference between a slice category and a coslice category.

First, recall the definitions.  Given a category \sf C, and an object A \in \mathrm{Obj}({\sf C}) in this category, the slice category {\sf C}_A is the category with

\mathrm{Obj}({\sf C}_A) := \{ (Z, f) \; | \; Z \in \mathrm{Obj}({\sf C}), f \in \mathrm{Hom}_{\sf C}(Z, A) \}

\mathrm{Hom}_{{\sf C}_A}((Z, f), (Y, g)) := \{\sigma \in \mathrm{Hom}_{\sf C}(Z, Y) \; | \; f = g \circ \sigma\}

In other words, objects of {\sf C}_A are objects of \sf C equipped with maps to A, and a map in {\sf C}_A is a map in \sf C which preserves this additional structure.  The coslice category is, of course, the categorical dual.

Now, before we move on, we need a slight generalization of this concept.  For some collection S of objects of C, define the generalized slice category (nonstandard terminology!) {\sf C}_S to be

\mathrm{Obj}({\sf C}_S) := \{ (Z, (f_A)_{A \in S}) \; | \; Z \in \mathrm{Obj}({\sf C}), \forall A \in S. f_A \in \mathrm{Hom}_{\sf C}(Z, A) \}

\mathrm{Hom}_S ((Z, (f_A)), (Y, (g_A)))
:= \{\sigma \in \mathrm{Hom}_{\sf C}(Z, Y) \; | \; \forall A \in S. f_A = g_A \circ \sigma \}

So, for instance, objects of {\sf C}_{A, B} are objects of \sf C equipped with maps to both A and B, and morphisms in this category are structure-respecting morphisms from \sf C. Generalized coslice categories, then, will be the same thing.

Now we can introduce a pair of facts which will prevent us from ever confusing slice and coslice categories again:

  • Products are terminal objects in generalized slice categories;
  • Coproducts are terminal objects in generalized coslice categories.

Bonus explanation: having trouble remembering the difference between products and coproducts?  Just think of them in the category of sets.  A product of sets is the usual cartesian product (together with projection maps) — hence the name — while a coproduct of sets is their disjoint union (together with inclusion maps), which is not something you’d generally call a product.

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