Motivation
There is a theory of smooth spaces (for example, diffeological spaces) as certain sheaves on the site $\mathrm{Cart}$, which is the "category of cartesian spaces" whose objects are $\{\mathbb{R}^n \mid n \in \mathbb{N}\}$ and whose morphisms are smooth maps. The site structure on $\mathrm{Cart}$ is often given by good open covers $\{U_i \hookrightarrow U\}$ such that each intersection $U_i \times_U U_j$ is either empty or diffeomorphic to $U$, as well as higher intersections $U_i \times_U U_j \times_U U_k \times_U \dots$. Supposedly, this means the (iterated) pullbacks of the diagram $\{U_i \hookrightarrow U\}$, which exist in $\mathrm{PSh}(\mathrm{Cart})$, are really (the Yoneda images of) pullbacks in $\mathrm{Cart}$. There's just one problem: pullbacks don't always exist, even for good open covers, because the empty set is not in $\mathrm{Cart}$! This causes some mild (but fixable) problems, which I won't discuss here.
The good news is, there's nothing to stop us from just adding $\varnothing$, to form a new category $\{\varnothing\} \cup \mathrm{Cart}$ (still with smooth maps as morphisms). The theory of (pre)sheaves on this category is completely the same as on $\mathrm{Cart}$, because such a presheaf $X \in \mathrm{PSh}(\{\varnothing\} \cup \mathrm{Cart})$ can be extended to simply send $\varnothing$ to the terminal object $* \in \mathrm{Set}$.
However, doing so raises an issue. The category $\mathrm{Cart}$ is important for another reason, namely as the Lawvere theory corresponding to $C^\infty$-rings. This is actually related to the setting of sheaves mentioned above; the Isbell dual of a $C^\infty$-ring is a diffeological space, and vice versa. If smooth spaces 'should' be defined on $\{\varnothing\} \cup \mathrm{Cart}$, then (if this duality means anything) so should $C^\infty$-rings. They should be defined as algebras over the ''initialised Lawvere theory'' $\{\varnothing\} \cup \mathrm{Cart}$, which are product-preserving functors $$\{\varnothing\} \cup \mathrm{Cart} \to \mathrm{Set}.$$ This makes the trivial $C^\infty$-ring $U \mapsto *$ corepresentable (by $\varnothing$). Of course, any $C^\infty$-ring gives rise to such a functor by sending $\varnothing$ to the initial object of $\mathrm{Set}$ (the empty set).
Another example is as follows. Fix a ring $R$; the Lawvere theory corresponding to $R$-algebras is $\{\mathbb{A}_R^n \mid n \in \mathbb{N}\}$, where $\mathbb{A}_R^n$ is the spectrum of $R[x]^{\otimes n}$. The added initial object is the spectrum of the zero $R$-algebra.
Question(s)
So, what's the deal with this? Does the theory of Lawvere theories differ much (or at all) if you add an initial object? Does anything break in their applications to universal algebra and logic? And if so, has anyone worked this all out?
A comment: adding an initial object to $\{T^n \mid n \in \mathbb{N}\}$ can be thought of as adding in $T^{-\infty}$. This idea comes from the degree function on polynomials, whose product-preserving property requires us to define the degree of the zero polynomial to be $-\infty$.
