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My original question is this. I found Zhen Lin's answer very useful, but I couldn't think of a category which is complete but has no initial object. The first category that I thought that has no initial object was the category of fields, but it isn't complete either. Researching for categories whitout initial object I found this math overflow post that shows the following examples

(1) A groupoid has an initial object precisely if it is equivalent to the trivial groupoid.

(2) Any disjoint union of two categories has no initial object.

(3) Any poset without smallest elements has no initial object.

The category of all grupoids is complete, but this isn't what example (1) intended to be (right?). And for (2) and (3) I couldn't decide completeness for both.

Could you help me with this examples, or even give a new example of a category as in the title?

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    $\begingroup$ You can find an example for (3) by taking a proper class of objects ordered so that every subset has a largest lower bound. (A set of objects would not suffice as their meet would be the smallest element.) For example, take the reversed order of all ordinals or cardinals. $\endgroup$
    – Berci
    Jan 11 at 8:12
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None of the examples of categories with no initial object you describe are complete. It is actually tricky to write down an example here. The difficulty is that by the adjoint functor theorem, a complete category $C$ has an initial object as soon as the terminal functor $T : C \to 1$ has a left adjoint, which is true once the terminal functor satisfies the solution set condition. This will be true in most common examples.

So let's see what it takes for this to not be true. The solution set condition for $T$ asks for there to exist a (small) family $c_i, i \in I$ of objects in $C$ indexed by a set $I$ such that every object $c \in C$ admits a morphism $c_i \to c$ for some $i$. This is a bit weaker than asking that the $c_i$ be a family of generators (which would involve asking for the morphisms to be epimorphisms); such a family $\{ c_i \}$ is a weakly initial family of objects. I believe the way the adjoint functor theorem works from here is that a suitable limit over this family is used to produce the initial object.

So to build a counterexample we need to find a complete category $C$ which does not admit a weakly initial family of objects. Since an initial object is in particular a weakly initial family of objects this will straightforwardly imply that $C$ does not have an initial object, but the point is that this stronger condition tells us where we need to look. We need a category containing objects which are so "large" that for any set-sized family of objects in the category there is some even larger object which is "inaccessible" relative to that family.

Berci's comment gives what seems to me to be two of the simplest examples of such a thing: take the opposite of the "large poset" of ordinals ordered by inclusion. To say that this category is complete but does not have an initial object is equivalent to saying that the large poset of ordinals ordered by inclusion is cocomplete (has all (small) joins / suprema) but does not have a terminal object (a largest element), both of which are true and standard properties of the ordinals. Similarly we can consider the "large poset" of sets ordered by inclusion, which again is cocomplete (equivalently, admits set-sized unions) but does not have a largest element (since there is no set of all sets).

In the setting of "large posets" it's not hard to see that the infimum of a weakly initial family of objects is an initial object, and dually that the supremum of a weakly terminal family of objects is terminal, so the equivalence between the two is clearer here.

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