Is the category of (pre)sheaves over a singleton isomorphic to the category of sets?

I’m wondering whether PSh({x}) or Sh({x}) are equivalent to the category of sets. For sheaves, since the value on an initial object is a terminal object, It seems like the only degree of freedom except choice of singleton is the value on {x}. Still though, I’m not sure whether I’m not talking nonsense. For presheaves, … Read more

Categories like FinSet\mathsf{FinSet}, but with elements of Z\mathbb{Z} or Q\mathbb{Q} as objects?

Suppose, we do universal algebra in a “non-evil” fashion, such that every algebra carries around an equivalence relation ≅ that poses as equality and such that size issues don’t matter (we may have proper classes as underlying “sets”). If we consider the category of finite sets with binary products ×, binary coproducts +, the empty … Read more

Confusion about categorical viewpoint of normal subgroups

One way to define normal subgroups is as normal monos in Grp. Another uses quotients by internal equivalence relations, which makes use (I think) of the fact Grp has effective equivalence relations – each equivalence relation is a the kernel pair of its coequalizer. What’s the theory linking these two definitions? Answer Let me state … Read more

If functions compose both ways to make automorphisms, are they isomorphisms?

Let’s say that we have morphisms f:A→B and g:B→A such that f∘g and g∘f are both automorphisms (an automorphism is a morphism that is both iso and endo). Are f and g isomorphisms? The converse is true. Also they won’t necessarily be inverses. If it is not true in general, which categories is it true … Read more

Why are concretizable categories locally small?

I have seen it mentioned in a few places that concrete (or concretizable) categories are locally small, but never seen any proof. Is it particularly trivial? If not, does anybody have some reference or sketch proof? Answer By definition, the concrete \mathcal C has a faithful functor U: \mathcal C \to \text{Set}. In other words, … Read more

How many adjunctions are there between (infinite) categories?

The title is a little bit imprecise. Consider “typical” categories A,B, let’s say infinite and locally small. A pair of adjointable functors (F,G) is a pair of functors (F:A→B,G:B→A), such that F is left-adjoint to G. We consider (F,G) and (F′,G′) the same, if F≅F′ or G≅G′. How many pairs of adjointable functors are there … Read more