Initial structures in the category of algebraic systems of the same type

Question

From Handbook of Analysis and Its Foundations by Eric Schechter

9.21. Basic properties of subalgebras. We consider the category consisting of the algebraic systems of some type $(τ, \mathcal{J})$, with homomorphisms of type $τ$. (For examples, think of the category of lattices, the category of rings, or the category of lattice groups.) In this category, a subobject is called a subalgebra. Let $X$ be an object.

$Y$ is a subalgebra of $X$ if and only if $Y$ is a subset of $X$ that is closed under the fundamental operations of $X$. It then follows that $Y$ itself is also an algebraic system of type $(τ, \mathcal{J})$, whose fundamental operations are the restrictions to $Y$ of the fundamental operations of $X$.

Except in degenerate cases, not every subset of $X$ is closed under the fundamental operations. Thus, in an algebraic category, initial structures do not always exist.

I was wondering how to understand why "in an algebraic category, initial structures do not always exist" is because "not every subset of $X$ is closed under the fundamental operations".

Is "initial structures" same as the concept of initial object in the category? Isn't the initial object the empty set? What does $\mathcal{J}$ mean? I know $\tau$ is the arity function and it itself can be called the type.

Thanks!

Answer 1

Perhaps the notation has been explained elsewhere in the book. My wild guess is that in $(\tau, \mathcal{J})$ the first symbol $\tau$ describes an algebraic signature, and the second symbol $\mathcal{J}$ denotes a class of logical sentences that have to be satisfied by algebras. For example, you may describe monoids as algebras over signature $\langle \circ/2, \epsilon/0\rangle$ satisfying sentences: $$\{x \circ (y \circ z) = (x \circ y) \circ z, \epsilon \circ x = x, x \circ \epsilon = x\}$$ An initial structure in (the category of) algebras of type $\tau$ satisfying $\mathcal{J}$ is by definition an initial object in this category.

An initial structure cannot be "the empty set", because the empty set is a set, and an initial structure has to be an algebra. However, in "most cases" the carrier of an initial structure is not the empty set either. For example in the category of monoids, the initial structure is the trivial monoid $\langle \{\epsilon\}, \circ, \epsilon \rangle$ (yes, it is also the terminal structure). Initial structures may not exist. For example, if we added an additional sentence: $$\exists_{x,y} \;x \not= y$$ to the above set of sentences characterizing monoids, then initial structures would not exist.

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Zhen is right, the definition refers to the concept of "lifting" defined here. The puzzling thing is that both your book and the referred page interchange the terminology --- what they call an initial lifting is really a terminal lifting, or cartesian lifting. And, yes, terminal liftings are terminal objects in appropriate categories.

The “right” terminology comes from fibrations --- if you pick $\Lambda = \{\bullet\}$, then your "initial" structures correspond to the terminal liftings in the underlying functor $U((X, \mathcal{T}) = X$ --- which is a fibration precisely when all $\{\bullet\}$-indexed "initial" structures exist.