underlying set of direct limit not the direct limit of underlying sets

Question

I am searching for a category $\mathcal C$ defined by a species of structures with morphisms $\Sigma$ (here I mean what is called 'espèce de structure' in Bourbaki Set Theory, chapter IV; put simply: the objects of $\mathcal C$ are sets endowed with some structure and morphisms are set-theoretic maps that respect this structure) such that the following holds:

• direct limits (taken over directed sets) exist in $\mathcal C$;
• the underlying set of a direct limit in $\mathcal C$ is not in general the direct limit of the underlying sets.

Please ask me if something is not clear. Thanks in advance.

Answer 1

The simplest examples of such things are categories of structures with infinitary operations.

For example, let $\mathcal{C}$ be the category of sets $X$ equipped with a function $\alpha : X^{\mathbb{N}} \to X$, with morphisms those maps that commute with $\alpha$. It is easy to see that $\mathcal{C}$ is complete, and in fact (but not so easily) $\mathcal{C}$ is also cocomplete.

I claim that the forgetful functor $\mathcal{C} \to \mathbf{Set}$ does not preserve filtered colimits in general. Indeed, if that were true, then the union of a chain of substructures would always be closed under $\alpha$, but one can straightforwardly find a counterexample. (For instance, take $X = \omega_1$ and $\alpha = \sup$.)

Answer 2

A typical example is the category of Banach spaces. The colimit of $\mathbb{C} \to \mathbb{C}^2 \to \mathbb{C}^3 \to \dotsc$ is $l^2(\mathbb{N})$. It is the completion of the colimit of the underlying normed spaces $\mathbb{C}^{\oplus \mathbb{N}}$.

A similar but more "discrete" example: The order of successor ordinals (considered as a category). The colimit of $1 \leq 2 \leq 3 \leq \dotsc$ is not $\omega$, but rather $\omega+1$.