I'm trying to find an example of a finite category in which there are no pushouts or pullbacks. By finite category, I mean a category with a finite amount of objects and morphisms.
The concepts of pushouts and pullbacks are new to me, but as far a I understand, they are similar to products and coproducts, only with a few more restrictions. So I tried to find a category with no products or coproducts, but I got nowhere. All of the examples of categories that I know (Gps, Top, Rng, Vect...) seem to have both, so I don't know how to get an example.
Could anyone please lend me a hand? Thanks in advance!
The easiest way of producing such examples is to look at some special cases of pullbacks and pushouts that can be better understood - products and coproducts. Constructing a category without products and coproducts (but with initial and terminal objects, to get them as pushouts and pullbacks) is quite easy just looking at sets. Take for example the full subcategory of $\mathrm{Set}$ with objects $\emptyset, \{0\}, \{0, 1\}$.
Note that products and coproducts in a subcategory do not need to coincide with products and coproducts from category itself, but in this case it is easy to check that both $\{0, 1\} \times \{0, 1\}$ and $\{0, 1\} \amalg \{0, 1\}$ must contain $4$ elements, thus do not exist in $\mathcal C$.
Less artificial, but not really finite (however you can just choose some arbitrary finite subcategory if you want) such category is for example a category of connected manifolds (together with empty set). The lack of coproducts is clear, so also the lack of pushouts. Even though products of connected manifolds are perfectly fine connected manifolds, this category do not have pullbacks in general. A simple counterexample can be constructed for instance from maps $f, g: \mathbb R \rightarrow \mathbb R$ $$ f(x) = |x(x-1)| \\ g(x) = 0 $$
The resulting pullback has a form $\{(x, y) \in \mathbb R^2 \ \ | \ \ f(x) = g(y)\}$, which is clearly not connected.