# The Unapologetic Mathematician

## Limits and Colimits

One of the big tools in category theory is the limit of a functor. In fact, depending on the source category, limits give whole families of useful constructions, including (multiple) equalizers, (multiple) products, (multiple) pullbacks, and more we haven’t talked about yet. The dual notion of a colimit similarly generalizes (multiple) coequalizers, (multiple) coproducts, (multiple) pushouts, and so on.

Given a functor $F:\mathcal{J}\rightarrow\mathcal{C}$ the limit of $F$ (if it exists) is a couniversal cone on $F$ — a terminal object in the comma category $(\Delta\downarrow F)$. That is, it consists of an object $L$ and arrows $\lambda_J:L\rightarrow F(J)$ for each object $J\in\mathcal{J}$, and for any other object $X$ with arrows $\xi_J:X\rightarrow F(J)$ there is a unique arrow $\eta:X\rightarrow L$ with $\xi_J=\lambda_J\circ\eta$ for all $J\in\mathcal{J}$. We often write $L=\varprojlim_\mathcal{J}F$ to denote the limit of the functor $F$ from the category $\mathcal{J}$. As usual, limits are unique up to isomorphism.

Dually, the colimit of $F$ (if it exists) is a universal cocone on $F$ — an initial object in the comma category $(F\downarrow\Delta)$. It’s given by an object $L$ and arrows $\lambda_J:F(J)\rightarrow L$ so that for any other object $X$ and arrows $\xi_J:F(J)\rightarrow X$ there is a unique arrow $\eta:L\rightarrow X$ with $\xi_J=\eta\circ\lambda_J$ for all $J\in\mathcal{J}$. We often write $L=\varinjlim_\mathcal{J}F$ to denote the colimit of the functor $F$ from the category $\mathcal{J}$.

If $\mathcal{J}$ is a set — a category with only identity arrows — then a functor from $\mathcal{J}$ to $\mathcal{C}$ is a collection of objects of $\mathcal{C}$, one for each element of $\mathcal{J}$. The limit of this functor is the product of these objects, and the colimit is their coproduct.

If $\mathcal{J}$ consists of two objects with a set of parallel arrows from one to the other, then a functor from $\mathcal{J}$ to $\mathcal{C}$ is a collection of parallel morphisms in $\mathcal{C}$. The limit of this functor is the equalizer of the collection, and the colimit is their coequalizer.

Check for yourself that limits over the category $\bullet\rightarrow\bullet\leftarrow\bullet$ are pullbacks, while colimits over the category $\bullet\leftarrow\bullet\rightarrow\bullet$ are pushouts.

If the category $\mathcal{J}$ has an initial object $I$ then we have a unique arrow $\iota_J:I\rightarrow J$ for every object $J\in\mathcal{J}$. Now for any functor $F:\mathcal{J}\rightarrow\mathcal{C}$ we see that $F(I)=\varprojlim_\mathcal{J}F$. Indeed, we can just use $F(\iota_J)$ for the required arrows, which are compatible with all the other arrows from $\mathcal{J}$ because $F$ is a functor. Any other cone on $F$ must have an arrow $\xi_I:X\rightarrow F(I)$, and compatibility requires that any other arrow $\xi_J:X\rightarrow F(J)$ in the cone is the composition $F(\iota_J)\circ\xi_I$. So if $\mathcal{J}$ has an initial object then limits are trivial.

The interesting case is when $\mathcal{J}$ does not have an initial object. We can add a new object to $\mathcal{J}$ with a unique arrow to every other object, getting a category $\overline{\mathcal{J}}$ which does have an initial object. The question is whether a given functor $F:\mathcal{J}\rightarrow\mathcal{C}$ can be extended to a functor $\overline{F}:\overline{\mathcal{J}}\rightarrow\mathcal{C}$. If it can, then the image of the new initial object is the vertex of a cone on $F$. If there is a couniversal such extension, that’s the limit of $F$.

Dually, if $\mathcal{J}$ has a terminal object $T$ then $F(T)=\varinjlim_\mathcal{J}F$. If $\mathcal{J}$ doesn’t have a terminal object, we can add one to get a category $\underline{\mathcal{J}}$, and colimits tell us when and how we can extend a functor $F:\mathcal{J}\rightarrow\mathcal{C}$ “universally” to a functor $\underline{F}:\underline{\mathcal{J}}\rightarrow\mathcal{C}$.

If there exists a limit in $\mathcal{C}$ for any functor from a finite category $\mathcal{J}$ we say that $\mathcal{C}$ “has finite limits”. If there exists a limit for any functor from a small category $\mathcal{J}$ we say that $\mathcal{C}$ is “complete”. The dual conditions are that $\mathcal{C}$ “has finite colimits” and that $\mathcal{C}$ is “cocomplete”, respectively.

June 19, 2007 - Posted by | Category theory

## 11 Comments »

1. […] Existence Theorem for Limits Of course, though we’ve defined limits, we don’t know in general whether or not they exist. Specific limits in specific categories […]

Pingback by The Existence Theorem for Limits « The Unapologetic Mathematician | June 20, 2007 | Reply

2. “The question is whether a given functor $F:\mathcal{J}\rightarrow\mathcal{C}$ can be extended to a functor $\overline{F}:\overline{\mathcal{J}}\rightarrow\mathcal{C}$. If it can, then the image of the new initial object is the limit of F.”

Are you sure that’s enough? That clearly shows that $\overline{F}$ has a limit, but F might have more cones. Actually the extension only gives us an arbitrary cone on F. (Or I have understood something incorrectly, which is very possible)

Comment by mattiast | June 22, 2007 | Reply

3. Ah, yes.. I oversimplified a bit there. Tweaking the entry now…

Comment by John Armstrong | June 22, 2007 | Reply

4. […] arrows and universal elements Remember that we defined a colimit as an initial object in the comma category , and a limit as a terminal object in the comma category […]

Pingback by Universal arrows and universal elements « The Unapologetic Mathematician | June 25, 2007 | Reply

5. […] a preorder, this adds nothing in that case. However, filtered categories show up in the theory of colimits in categories. In fact originally colimits were only defined over filtered index categories […]

Pingback by Nets, Part I « The Unapologetic Mathematician | December 10, 2007 | Reply

6. […] if given on , assigns taken over sets containing . If you don’t know limits, check out John Armstrong’s definition, but generally the inverse image sheaf isn’t very helpful, later we’ll discuss a more […]

Pingback by Morphisms of Sheaves « Rigorous Trivialities | January 30, 2008 | Reply

7. Please, can you help me prove that if I is a small category and the identity functor of I has colimit, then his colimit is final object in I? I have an exam tomorrow:(

Comment by Lucia | February 2, 2009 | Reply

8. […] inverse limit construction is perhaps best thought of in more generality, as an example of a categorical limit. Then inverse limits can be defined in arbitrary categories, although they need not exist.  […]

Pingback by Generalities on completions « Delta Epsilons | August 24, 2009 | Reply

9. […] , , and so on. We can even define the infinitely differentiable functions to be the limit (in the categorical sense) of this process. It consists of all the functions that are in for all natural numbers . Taking […]

Pingback by Smoothness « The Unapologetic Mathematician | October 21, 2009 | Reply

10. […] viewpoint also brings us into contact with the category-theoretic notion of a colimit (feel free to ignore this if you’re category-phobic). Indeed, if we define […]

Pingback by Algebras of Sets « The Unapologetic Mathematician | March 15, 2010 | Reply

11. […] answer is to use the categorical definition of a limit. Given a point the collection of open neighborhoods of form a directed set, and we can take the […]

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