Why are equivalence relations essential for definitions of colimits but not for those of limits?

Question

In category theory, equivalence relations are essential to define colimits (pushouts, coequaliziers, etc.), while definitions of limits (pullbacks, equalizers, etc.) include no equivalence relations. So, can one explain why equivalence relations are necessary for colimits but not for limits?

Answer 1

First of all, equivalence relations are not necessary to define colimits. The notion of a colimit (or limit, for that matter) is completely independent. A colimit of a diagram is a universal cocone, nothing more.

You are asking about the construction of colimits in specific categories. Let me explain the two differences:

1. The construction is not identical to the definition of something. The definition can be really abstract, whereas the construction must be concrete, often as concrete as possible in order to work with it. For example, you can define the completion of a metric space via its universal property, but its construction is done via equivalence classes of Cauchy sequences. Also here, there is no equivalence relation in the definition of the completion of a metric space.

2. The notion of a limit is completely dual to the notion of a colimit. Thus, when it comes to general constructions, there cannot be any difference. For example, it is an easy observation that if $T$ is a monad on a category $\mathcal{C}$, then the forgetful functor $\mathrm{Alg}(T) \to \mathcal{C}$ creates limits. This applies to a lot of forgetful functors in practice (for example, $\mathbf{Grp} \to \mathbf{Set}$). The dual statement is that for every comonad $T$ on a category $\mathcal{C}$ the forgetful functor $\mathrm{CoAlg}(T) \to \mathcal{C}$ creates colimits. (For example, colimits of $k$-coalgebras can be built directly from colimits of $k$-modules.) So as for the general theory, there cannot be any difference between limits and colimits. The differences only appear when you look at specific examples of categories, or more generally, classes of categories which are not closed under dualization (for example, the class of locally presentable categories).

It is true that in many concrete categories, the construction of limits does not require any equivalence relations, but the construction of colimits does. For example, this is the case for all algebraic categories. But: for every such example, we can take the dual category, in which the construction of limits does require equivalence relations, but the construction of colimits does not.

Take $\mathbf{Set}^{\mathrm{op}}$ for example. Here, limits typically involve equivalence relations. This category is actually equivalent to the category $\mathbf{CABA}$ of complete atomic boolean algebras (since every such algebra is isomorphic to the powerset $P(X)$ of some set $X$, the set of atoms, and every morphism $P(X) \to P(Y)$ is induced by a unique map $Y \to X$). With this representation, you will not be able to see any dualization anymore. It is a very concrete category. But still, limits behave much differently than what you are used in algebraic categories. For example, the coproduct of $P(X)$ and $P(Y)$ is $P(X \times Y)$, whereas the product of $P(X)$ and $P(Y)$ is $P(X \sqcup Y)$.

Finally, let me explain how the constructions of limits and colimits are similar, even in general:

An important result regarding the construction of limits is the following: Say we have a diagram $X=(X_i)_{i \in \mathcal{I}}$ in a category $\mathcal{C}$. Assume that the products $\prod_{i} X_i$ and $\prod_{i \to j} X_j$ exists. Then there are two canonical morphisms $\alpha,\beta : \prod_{i} X_i \rightrightarrows \prod_{i \to j} X_j$, defined by $p_{i \to j} \circ \alpha = p_j$ and $p_{i \to j} \circ \beta = X_{i \to j} \circ p_i$. Then, a limit of $X$ is the same as an equalizer of $\alpha,\beta$.

In other words (when the products and equalizers above exist): The limit can be constructed as a subobject of the product $\prod_i X_i$. Using the language of generalized elements, it consists of those elements $(x_i)$ such that $X_{i \to j}(x_i) = x_j$ for all $i \to j$. We restrict the elements.

The dual result states that a colimit of $X$ can be constructed as a quotient of the coproduct $\coprod_{i} X_i$, namely as a coequalizer of two morphisms $\coprod_{i \to j} X_i \rightrightarrows \coprod_{i} X_i$. But instead of restricting the elements, we need to identify certain elements.

Again: there cannot be a real difference, due to dualization, but: the notion of a generalized element is not self-dual. So when it comes to the description of elements, differences between limits and colimits will occur.

The description of colimits above would be completely identical to the description of limits if we used "coelements" (don't use this word, I just made it up), which are morphisms to a specific test object. Namely, coelements of the colimit of $X$ are special coelements of the coproduct $\coprod_i X_i$.

For example, we know that any (generalized) element of $\prod_{i} X_i$ yields a family of elements of the $X_i$. But this is far from being true for $\coprod_i X_i$. Even worse: not every element of $\coprod_i X_i$ must be induced by an element of $X_i$ for some index $i$. (Think about abelian groups.) In contrast, a coelement of $\coprod_i X_i$ is just a family of coelements of the $X_i$.

Let me finish with a specific example. Consider the two maps $\alpha,\beta : \mathbb{Z} \to \mathbb{Z}$ defined by $\alpha(z) := z$ and $\beta(z) := z+2$. The coequalizer of $\alpha,\beta$ in the category $\mathbf{Set}$ is the universal morphism $q : \mathbb{Z} \twoheadrightarrow Q$ with $q(z)=q(z+2)$ for all $z \in \mathbb{Z}$. Even though the general construction of a coequalizer in $\mathbf{Set}$ involves equivalence relations, we do not need one here. We can explicitly define $Q := \{0,1\}$, $q(z) := 0$ if $q$ is even and $q(z) := 1$ otherwise. Also, the universal property of $Q$ (or, more precisely, $q$) states that for every set $T$ the maps $\overline{f} : Q \to T$ correspond bijectively to the maps $f : \mathbb{Z} \to T$ with $f(z)=f(z+2)$ for all $z \in \mathbb{Z}$, via $f = \overline{f} \circ q$. This describes the coelements of $Q$ via special coelements of $\mathbb{Z}$ (very much like the equalizer is given by special elements!). Since the coequalizer is defined by this universal property, this definition does not involve any equivalence relations.