Limits in functor categories

Today I want to give a great example of creation of limits that shows how useful it can be. For motivation, take a set $X$ , a monoid $M$ , and consider the set $M^X$ of functions from $X$ to $M$ . Then $M^X$ inherits a monoid structure from that on $M$ . Just define $\left[f_1f_2\right](x)=f_1(x)f_2(x)$ and take the function sending every element to the identity of $M$ as the identity of $M^X$ . We’re going to do the exact same thing in categories, but with having limits instead of a monoid structure.

As a preliminary result we need to note that if we have a set of categories $\mathcal{C}_s$ for $s\in S$ each of which has $\mathcal{J}$ -limits, then the product category $\prod\limits_{s\in S}\mathcal{C}_s$ has $\mathcal{J}$ -limits. Indeed, a functor from $\mathcal{J}$ to the product consists of a list of functors from $\mathcal{J}$ to each category $\mathcal{C}_s$ , and each of these has a limiting cone. These clearly assemble into a limiting cone for the overall functor.

The special case we’re interested here is when all $\mathcal{C}$ are the same category. Then the product category $\prod\limits_S\mathcal{C}$ is equivalent to the functor category $\mathcal{C}^S$ , where we consider $S$ as a discrete category. If $\mathcal{C}$ has $\mathcal{J}$ -limits, then so does $\mathcal{C}^S$ for any set $S$ .

Now, any small category $\mathcal{S}$ has a discrete subcategory $|\mathcal{S}|$ : its set of objects. There is an inclusion functor $i:\left|\mathcal{S}\right|\rightarrow\mathcal{S}$ . This gives rise to a functor $\mathcal{C}^i:\mathcal{C}^\mathcal{S}\rightarrow\mathcal{C}^{|\mathcal{S}|}$ . A functor $F:\mathcal{S}\rightarrow\mathcal{C}$ gets sent to the functor $F\circ i:\left|\mathcal{S}\right|\rightarrow\mathcal{C}$ . I claim that $\mathcal{C}^i$ creates all limits.

Before I prove this, let’s expand a bit to understand what it means. Given a functor $F:\mathcal{J}\rightarrow\mathcal{C}^\mathcal{S}$ and an object $S\in\mathcal{S}$ we can get a functor $\left[F(\underline{\hphantom{X}})\right](S):\mathcal{J}\rightarrow\mathcal{C}$ that takes an object $J\in\mathcal{J}$ and evaluates $F(J)$ at $S$ . This is an $|\mathcal{S}|$ -indexed family of functors to $\mathcal{C}$ , which is a functor to $\mathcal{C}^{|\mathcal{S}|}$ . A limit of this functor consists of a limit for each of the family of functors. The assertion is that if we have such a limit — a $\mathcal{J}$ -limit in $\mathcal{C}$ for each object of $\mathcal{S}$ — then these limits over each object assemble into a functor in $\mathcal{C}^\mathcal{S}$ , which is the limit of our original $F$ .

We have a limiting cone $\lambda_{J,S}:L(S)\rightarrow[F(J)](S)$ for each object $S\in\mathcal{S}$ . What we need is an arrow $L(s):L(S_1)\rightarrow L(S_2)$ for each arrow $s:S_1\rightarrow S_2$ in $\mathcal{S}$ and a natural transformation $L\rightarrow F(J)$ for each $J\in\mathcal{J}$ . Here’s the diagram we need:
Pointwise Limits Diagram

We consider an arrow $j:J_1\rightarrow J_2$ in $\mathcal{J}$ . The outer triangle is the limiting cone for the object $S_1$ , and the inner triangle is the limiting cone for the object $S_2$ . The bottom square commutes because $F$ is functorial in $\mathcal{S}$ and $\mathcal{J}$ separately. The two diagonal arrows towards the bottom are the functors $F(J_1)$ and $F(J_2)$ applied to the arrow $s$ . Now for each $J$ we get a composite arrow $\left[F(J)\right](s)\circ\lambda_{J,S_1}:L(S_1)\rightarrow\left[F(J)\right](S_2)$ , which is a cone on $\left[F(\underline{\hphantom{X}})\right](S_2)$ . Since $\lambda_{J,S_2}:L(S_2)\rightarrow\left[F(J)\right](S_2)$ is a limiting cone on this functor we get a unique arrow $L(s):L(S_1)\rightarrow L(S_2)$ .

We now know how $L$ must act on arrows of $\mathcal{S}$ , but we need to know that it’s a functor — that it preserves compositions. To do this, try to see the diagram above as a triangular prism viewed down the end. We get one such prism for each arrow $s$ , and for composable arrows we can stack the prisms end-to-end to get a prism for the composite. The uniqueness from the universal property now tells us that such a prism is unique, so the composition must be preserved.

Finally, for the natural transformations required to make this a cone, notice that the sides of the prism are exactly the naturality squares for a transformation from $L$ to $F(J_1)$ and $F(J_2)$ , so the arrows in the cones give us the components of the natural transformations we need. The proof that this is a limiting cone is straightforward, and a good exercise.

The upshot of all this is that if $\mathcal{C}$ has $\mathcal{J}$ -limits, then so does $\mathcal{C}^\mathcal{S}$ . Furthermore, we can evaluate such limits “pointwise”: $\left[\varprojlim_\mathcal{J}F\right](S)=\varprojlim_\mathcal{J}(F(S))$ .

As another exercise, see what needs to be dualized in the above argument (particularly in the diagram) to replace “limits” with “colimits”.

June 23, 2007 - Posted by John Armstrong | Category theory

9 Comments »

wow, your blog is all math. this is very much an interest to you isn’t it? are you a professor? and are you from maryland?

Comment by Hallie Honey | June 23, 2007 | Reply
I grew up a long time in Maryland and did my undergraduate work at College Park. I completed my Ph.D. at Yale, and yes, I’ll be a professor in the math department at Tulane in the fall. Until then I’m hanging around central Maryland where my parents still live.

Comment by John Armstrong | June 23, 2007 | Reply
Hi John,

Under suitable assumptions on your category C (e.g., if C has arbitrary *coproducts* (!)), there’s another way of deriving this result, using the fact that the functor C^S –> C^|S| is monadic, hence preserves and reflects (creates) any limits which happen to exist. (I haven’t checked; have you talked about the Eilenberg-Moore category of algebras somewhere on your blog?) Similarly, if C has arbitrary products, then C^S –> C^|S| is comonadic.

Comment by Todd Trimble | June 24, 2007 | Reply
No, I haven’t gone into Eilenberg-Moore, nor monads. In fact, I haven’t quite talked about monoidal categories yet, which would be a precursor (in my mind) to that sort of thing. It’s a good point, though.

Comment by John Armstrong | June 24, 2007 | Reply
[…] the proof of this is so similar to how we established limits in categories of functors, I’ll just refer you back to that and suggest this as practice with those […]

Pingback by Taking limits is functorial « The Unapologetic Mathematician | June 24, 2007 | Reply
Lots of new non-parsing formulas in this one

Comment by Avery | January 8, 2008 | Reply
If I is a small category ans it has an initial object, i, how can I prove that any functor F:I->C has limit and the limit is F(i) ?

Comment by Lucia | February 2, 2009 | Reply
Lucia, I am not in the business of doing your homework.

Comment by John Armstrong | February 2, 2009 | Reply
Thank you 🙂 you were very helpful:)

Comment by Lucia | February 2, 2009 | Reply

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