Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares about, and thus why he was thinking in this direction, but I am now wondering;
As working mathematicians, what category are you concerned with that does not have an initial object?
I am sorry if this question is slightly strange, I have made it CW because it seems appropriate.
Thanks!
Isn't it so that the category of fields doesn't have an initial object? I seem to recall seeing a lot of F_uss about this on the web from time to time.
I guess the question is meant to ask abut "classical concrete categories" only, those whose objects are sets with structure. For otherwise, the class of examples that we certainly care about is vast and uninteresting (as a class).
For instance a groupoid has an initial object precisely if it is equivalent to the trivial groupoid.
Any disjoint union of two categories has no initial object.
Any poset without smallest elements has no initial object.
The bordism categories which arise in the study of topological quantum field theories are extremely interesting but don't have initial or terminal objects (except in degenerate cases). These categories have some flavor of d-dimensional manifolds for objects and (d+1)-dimensional cobordisms between them for morphisms. See the Wikipedia and n-lab articles for further details and references.
Any programming languages based on lambda calculus (Haskell, (OCa)ML, Clean, Coq, Agda) forms a category -- in many different ways, in fact. One way is to have an object for each type of the language and a morphism for each well-typed expression with exactly one free variable. If the programming language has linear types then this category will not have a terminal object, since such an object would imply the ability to "discard" a value of any type.
Linear types are extremely useful in giving pure functional programming languages the ability to express side-effecting operations. Loosely, you have a linear type called "world"; every impure function both takes and returns a value of that type (and perhaps other values too). Since you can't duplicate or discard a "world", its handling imposes a deterministic evaluation order on all of the impure functions, and program transformations will not alter this order. The aggregate result of the side effects can then be reasoned about (formally, even).
The category of varieties (or just integral schemes) has no initial object. (The empty set is not irreducible; this does not prove my claim, but indicates that the naive choice for an initial object does not work).
In universal algebra and model theory, one usually requires that the underlying set be non-empty. If the signature has constants in it, then this is no restriction, but otherwise, there is no initial object. For example, under this definition there is no initial object in the category of semigroups. The reason for this restriction is that bad things happen to first-order logic when the underying carrier set is empty. Many standard theorems break down when you allow the empty domain. For example, the following theorem of first-order logic $$ \forall x P(x) \rightarrow \exists x P(x) $$ becomess false.
The category of fields has no initial object: It has one connected component for each characteristic, each of which has an initial object (this is a special case of Urs Schreiber's general remark about non-connected categories, but one I care about...)
Let R be the hyperfinite type III1 factor (if you don't know what that is, "let R be a ring" is a good enough approximation). The following category is equivalent to the category of R-R-bimodules on infinite dimensional separable Hilbert spaces.
Objects: Unital ring homomorphisms R → R.
Morphisms: Hom(φ,ψ) =
{x∈R : ∀y∈R, x φ(y) = ψ(y) x }
Composition of morphisms is given by multiplication in R. This category is actually a strict monoidal, with monoidal structure given by composition of ring homomorphisms.
The category of CDG-rings (curved DG-rings) does not have an initial object. (A CDG-ring B = (B,d,h) is a graded ring B endowed with an odd derivation d of degree 1 and an element h of degree 2 such that d2(b) = [h,b] for any b from B and d(h)=0. A morphism of CDG-rings (B,d_B,h_B) → (A,d_A,h_A) is a pair (f,a), where f is a morphism of graded rings A → B and a is an element of A of degree 1 such that f(d_B(b)) = d_A(f(b)) + [a,f(b)] for any b from B and f(h_B) = h_A + d_A(a) + a2, where [,] denotes the supercommutator (with signs). I leave the definition of the composition of morphisms to the readers of this answer.)