TITLE: Examples of a categories without products
QUESTION [25 upvotes]: A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category of fields. But I cannot formally prove why categorical product does not exist in these categories.
I can see that category of finite fields cannot have infinite products due to cardinality reasons, but I would like to know why category of fields cannot even have finite products.
Is there any other simple example of a category without (finite) products? Any help regarding this appreciated.

REPLY [12 votes]: Posets (viewed as categories) provide many examples of categories without certain limits. A product of two elements in a poset is simply a greatest lower bound, so you just need to make a poset where some pair of elements has no greatest lower bound. For example, the poset $\{ a, b, c, d \}$ where $a \leq c, a \leq d, b \leq c, b \leq d$. Then both $a$ and $b$ are lower bounds for $c$ and $d$, but they are incomparable.