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What are some examples of cocomplete categories without equalizers? And what are some examples of cocomplete categories without binary products?

They must exist, but at the moment I don't know any. Of course, you may also list examples of complete categories without coequalizers (resp., binary coproducts).

In some sense, cocomplete categories that are not complete have to be "quite large". For example they cannot have a small dense subcategory (MO/45577). I am quite sure that we can construct artificial examples, but I prefer to see examples that are (weird) full subcategories of commonly studied categories. I am only interested in $1$-categories here.

Some examples of weird cocomplete categories: $(\mathbf{On},\leq)$ is cocomplete, but it has no terminal object. But that one has binary products, too, and for trivial reasons also equalizers. $\mathbf{Met}_c$ (metric spaces with continuous maps) is cocomplete, but it doesn't have uncountable products (MSE/139168). But it has countable products and equalizers.

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    $\begingroup$ As I discussed in an answer here, Bezhanishvili and Kornell showed that in the category $\mathrm{Top}_{\mathrm{open}}$ of topological spaces with open maps, the product of Sierpiński space with itself fails to exist. Is this category cocomplete? $\endgroup$ Commented 3 hours ago
  • $\begingroup$ @JamesEHanson Yes Todd has proven there that it is cocomplete. It should be noted that both Todd and the authors of the paper actually mean continuous open maps. It seems that some authors assume this, but I find this is a mistake, it causes unnecessary confusion. See also Wikipedia. $\endgroup$ Commented 2 hours ago

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