I always have trouble remembering the details of categorical constructions. (Do colimits commute with left or right adjoints? Left. Are limits universal for maps into our out of a diagram? Into.) Hopefully, by organizing the details and writing them all down in one place, I’ll have a chance to finally get them straight. I plan to revise this post over time.

1. (co)limits

A category is said to be small if the collection of objects and the collection of morphisms are sets. A functor is called a diagram index by .

Definition 1 Let be a diagram index by . The limit of , if it exists, is an object together with maps maps for each such that, for each we have , and is universal for maps into . In particular, that means if there is an object and maps satisfying for each , there is unique satisfying .

The limit is also called the inverse limit or projective limit. However, I can never keep these names straight.

Let’s look at some examples.

Example 1 Let denote the category with the objects and no morphisms. Then the limit of a functor is just the product

Example 2 If is the empty category, then the limit of is the final object.

Example 3 The kernel is the limit of a diagram indexed by the diagram .

The colimit is dual construction, obtained by reversing all the arrows. Specifically,

Definition 2 Let be a diagram index by . The colimit of , if it exists, is an object together with maps maps for each such that, for each we have , and is universal for maps out of . In particular, that means that if there is an object and maps satisfying for each , then there is unique satisfying .

The colimit is also called the injective limit.

Example 4 The colimit of a diagram is the coproduct . In particular, the colimit of is the initial object.

Example 5 The cokernel is the colimit of a diagram indexed by .

The (co)kernel is a good example to remember that kernels, and therefore limits, map into, and cokernels and tehrefore colimits are mapped into.

2. Adjoints

Definition 3 A pair of functors and are adjoint if, for each and , there are bijections and the are natural. More precisely, that means is a natural transformation of bifunctors. Unrolling definitions, that means for all , the following diagram commutes where is the pullback of functions, and for each the following diagram commutes where is the pushforward of functions.

We say is left adjoint to and is right adjoint to .

Example 6 The classical example of adjoint functors is tensor product and in, say, abelian groups. is left adjoint to via the natural isomorphisms where is bilinear maps .

There are tons of examples of adjoint functors. We’ll see more examples when we get to operations on sheaves.

Finally, I’ll conclude with some key (which I won’t explain but may elaborate on later).

Theorem 4 Limits commute with limits and right adjoints. To make the latter precise, if is a right adjoint and a diagram indexed by , then . In particular, since the kernel is a limit, limits and right adjoints are left exact. Similar, colimits commute with colimits and left adjoints. In particular, since the cokernel is a colimit, colimits and left adjoints are left exact.

Does anyone have a nice mnemonic to remember this?

Finally, I’d be remiss if I didn’t mention that much of this presentation comes from Ravi Vakil’s notes, Foundations of Algebraic Geometry. I learned about this stuff from a few different sources, but his notes were the most useful and contain lots of great exercises.

Next time, we’ll get back to sheaves.