5.4. Preservation of Limits

Let \(F: J \to C\) be a diagram and suppose \(G: \cc \to \dd\) is a functor. If for every limit \(\Lim F\) exists in \(\cc\) with morphisms \(u_i: C \to F_i\), we say \(G\) preserves limits if \(G(\Lim F)\) is a limit with morphisms \(G(u_i): G(C) \to G(F_i)\). Moreover, we call such a functor a continuous functor.

As an immediate consequence of the definition, it should be noted that a composition of continuous functors is continuous.

Below we see a visual definition of a continuous functor.

There's one particular and important functor which is always continuous in any category.

Let \(\cc\) be a small category. Then for each \(C \in \cc\), the functor

\[ \hom_{\cc}(C, -): \cc \to **Set** \]

preserves limits. (Dually, the functor \(\hom_{\cc}(-, C) = \hom_{\cc}(C, -): \cc\op \to **Set**\) takes colimits to limits.)

Let \(F: J \to \mathcal{C}\) be a diagram with a limiting object
\(\text{Lim } F\) equipped with the morphisms \(\sigma_i: \text{Lim } F \to F_i\). Then applying the \(\text{Hom}_{\mathcal{C}}(C, -)\) functor to \(\text{Lim } F\) and to each \(u_i\), we realize it forms a cone in \(**Set**\).

Now we show that \(\text{Hom}_{\mathcal{C}}(C, \text{Lim } F)\), equipped with the morphisms \(\sigma_{i*}\), is a universal cone; that is, it is a limit. Suppose that \(X\) is a set which forms a cone with the morphisms \(\tau_i: X \to \text{Hom}_{\mathcal{C}}(C, F_i)\).

Then for each \(x \in X\), we see that \(\tau_i(x) : C \to F_i\). The diagram above tells us that \(u \circ \tau_i(x) = \tau_j(x)\) for each \(x\). Hence each \(x \in X\) induces a cone with apex \(C\) with morphisms \(\tau_i(x): C \to F_i\).

However, \(\text{Lim } F\) is the limit of \(F: J \to \mathcal{C}\). Therefore, there exists a unique arrow \(h_x: C \to \text{Lim } F\) such that \(h_x \circ \sigma_i = \tau_i(x)\). Now we can uniquely define a function \(: X \to \text{Hom}_{\mathcal{C}}(C, \text{Lim } F)\) where \(h(x) = h_x: C \to \text{Lim } F\), in such a way that the diagram below commutes.

Therefore, \(\text{Hom}_{\mathcal{C}}(C, \text{Lim } F)\) is a limit in Set. At this point, you may be wondering: What is the difference between a functor which "creates limits" and one which preserves them? We'll see that their definitions are different, but creating limits is the same as preserving them

Suppose \(G: \cc \to \dd\) creates limits for \(F: J \to \cc\). If \(G \circ F: J \to \dd\) has a limit in \(\dd\), then \(G\) is continuous.

Suppose \(F: J \to \cc\) has limit \(\Lim F\) in \(\cc\) with morphisms \(v_i: \Lim F \to F_i\) for each \(i \in J\). Further, suppose \(G \circ F: J \to \dd\) has a limit \(\Lim G \circ F\) with morphisms \(u_i: \Lim G \circ F \to G\circ F_i\).

Since \(G: \cc \to \dd\) creates limits, this implies the existence of a limiting object \(X\) with morphisms \(\sigma_i: X \to F_i\) for \(F: J \to C\) where \(G(X) = \Lim G\circ F\) and \(G(\sigma_i) = u_i\). However, limiting objects are unique (by their universal properties).
As they must be isomorphic, there exists an isomorphism \(\phi: X \to \Lim F\) for which \(v_i \circ \phi= \sigma_i\). Thus we see that

\[ G(\Lim F) \cong G(X) = \Lim G \circ F \qquad G(v_i \circ \phi) = G(\sigma_i) = u_i. \]

Therefore, \(G\) preserves limits and so is continuous.

We have the following as a corollary.

Suppose \(G: \cc \to \dd\) creates limits and \(\cc\) is complete. Then \(\dd\) is complete and \(G\) preserves limits.