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No such category exists. My original argument for this assumed local smallness and is below the break; here is a simpler argument that does not require local smallness (though it does basically use m …

You can recover $F$ from $\mathrm{Res}_F$; this is an exercise in using the Yoneda lemma. Let $H_A$ denote the presheaf represented by an object $A$ of $\mathbf C$ or $\mathbf D$. Then for $A\in \ma …

For any $n$, there is an $n$-ary smooth function that is not a composition of smooth functions of lower arity; according to this answer to a very similar question, this is due to Vitushkin (at least f …

Here's a very simple example. Let $\mathcal{C}$ be any set with more than two elements, considered as a discrete category. Then $\mathcal{C}=\mathcal{C}^{op}$, and any permutation $F:\mathcal{C}\to\ …

An additive category need not be balanced. Consider the full subcategory of abelian groups consisting of all torsion-free groups. Then for any $n\neq0$, the map $n:\mathbb{Z}\to\mathbb{Z}$ is monic …

I'm not sure this is quite what you're looking for, but if A is small, you can consider the (contravariant) Yoneda embedding of A into the category of left-exact functors from A to Ab. This is an exa …

Yes, certainly. For instance, you could define $F(t)=a$ for $a<1$ and $F(1)=b$, with the obvious choice of morphisms (the identity whenever possible, and otherwise $m$). More generally, if $I=A\cup …

Sure, they come naturally from looking at higher-order rearrangements of products. For example, the pentagon axiom is exactly saying that there is only one way to go between any two ways of parenthes …

Here's a nice example that recently came up in an MSE question. Let $k$ be a field and let $Vect$ be the category of $k$-vector spaces and $Aff$ be the category of $k$-affine spaces. Every vector sp …

Consider the category consisting of that diagram, together with an extra object $P$ with maps $Y\leftarrow P\to C$ that commute with the maps to $Z$. Then in this category, the whole diagram and the …

I would disagree that the hypotheses of the adjoint functor theorem are much stronger than exactness. Left exactness is equivalent to preserving all finite limits, and the hypotheses of the adjoint f …

When you pass to a larger universe, all categories that were in your old universe become small. A small category which is not a poset cannot be closed under colimits (think about taking coproducts wi …

It is trivial that such a maximal $E$ exists, and consists of pairs $\{f,g\}$ such that whenever $x$ and $y$ are adjacent, $f(x)$ and $g(y)$ are adjacent. However, it does not make $\operatorname{Hom …

You can define it the same way as you define a direct limit of sets, using Scott's trick to form equivalence classes. Whenever you have an equivalence relation defined on a class $X$, you can form eq …

The empty category trivially satisfies this (there are no functors at all from a nonempty category to the empty category), but no other such category exists. Let $A$ be any category with a terminal o …