# Limit Construction in Category Theory

Consider $A\xrightarrow{ \ f \ }B$, be a mapping in the category of sets. I would like to get a glimpse of how one argues in category theory with regards to finding the limit of the above diagram.

I understand that a limit is a left cone $C$ and two maps $C\xrightarrow{ \ c_1 \ }A$ and $C\xrightarrow{ \ c_2 \ }B$ (satisfying $c_2 = f \circ c_1$) which is terminal, i.e., for any other cone $\tilde{C}$, there exists unique morphism from $\mu: \tilde{C}\to C$, which makes everything commute.

My first question is the following:

How does one go about constructing $C$, $c_1 , c_2$ in the first place? Everything feels terribly abstract. My gut feeling is that we should construct $C$, only from the given data, namely $A\xrightarrow{ \ f \ }B$, and the only "natural thing" for me would be $C=A$, $c_1=\mathbb{1}_A , c_2=f$, as this is effectively the only givens that I have.

Is the whole point of category theory to make a complicated construction once and then by the universal property argue that this is the only construction available (up to isomorphisms in the category)?

My second question is with regards to a similar construction of the pullback:

If I am given $f:A\to W$ and $g:B\to W$, then a limit of this diagram would be some set $C$ and a pair of maps $c_1 , c_2$ that makes everything commute and is furthermore a terminal cone. Could I see please a clear construction and explanation of why $C$ should be $A\times_W B$? Does this construction require of the category to have equalizers?

How does one go about constructing $C$, $c_1 , c_2$ in the first place?

Your answer for the limit of $A\xrightarrow{ \ f \ }B$ is correct, as is the intuition you used to arrive at it. If however you would like a more methodical answer, there is a general construction for concrete limits over directed systems. It tells you that the limit is $\{(a,b)\in A\times B\mid f(a)=b\}.$ This set is called the graph of $f$, and it is isomorphic to $A$.

Could I see please a clear construction and explanation of why $C$ should be $A\times_W B$?

Well $A\times_W B$ makes the diagram commute, and it imposes only those equations required for making it commute.

Does this construction require of the category to have equalizers?

If a category has all pullbacks and a terminal object, then it has all equalizers as well (since an equalizer may be written as a pullback). But it is possible for a category to have one pullback, and fail to have some other equalizer.

• It's not quite true that an equalizer can be written as a pullback. The usual construction requires also having products which having just pullbacks does not imply. However, if you have a terminal object, then having pullbacks implies having products and thus equalizers. Of course, Set has all these things. – Derek Elkins Nov 4 '17 at 14:12
• @DerekElkins thank you for the correction. – ziggurism Nov 4 '17 at 14:15

In the category of sets, there's an easy construction of the limits $L$ of diagrams $D$. The elements of $L$ are in obvious bijection with functions $1\to L$, where $1$ is your favorite one-element set. By definition of limit, these functions are in natural bijection with the cones from $1$ to $D$. Such a cone is a family of functions, $1\to X$ for all the objects $X$ in $D$, subject to commutativity conditions (as in the definition of "cone"). But a function $1\to X$ amounts to an element of $X$, so you have a family of "chosen" elements, one in each set $X$ of the diagram $D$. The commutativity requirements for a cone then say that, if $f:X\to Y$ is any one of the morphisms in $D$, then $f$ has to map the chosen element of $X$ to the chosen element of $Y$. Putting all of this together, you get that an element of $L$ amounts to a choice of particular elements from all the sets in $D$, such that the chosen elements map to each other under all the morphisms in $D$. (Note that, when $D$ is just $f:A\to B$, this description reduces to the $(a,f(a))$ description in ziggurism's answer, and that the general description also reduces to the explicit description of pullbacks.)