nLab categorical properties of Frölicher spaces

Idea

On this page we shall record those aspects of the theory of Frölicher spaces that are particularly categorical in nature.

Colimits

Proposition

Every Frölicher space is functorially the colimit of a diagram of manifolds. In fact, it is a colimit of a diagram in the full subcategory consisting of the single object $\mathbb{R}$.

Proof

Let $(X,C,F)$ be a Frölicher space. Let $\mathcal{C}$ be the category whose objects are the elements of $C$ and the morphisms $c \to c'$ correspond to the smooth functions $g \colon \mathbb{R} \to \mathbb{R}$ with $c' \circ g = c$. Note that for a fixed curve $c'$ and a smooth function $g \colon \mathbb{R} \to \mathbb{R}$ then from the definition of a Frölicher space there is a curve $c \in C$ such that $g$ defines a morphism $c \to c'$ (take $c = c' \circ g$).

Define a functor $G \colon \mathcal{C} \to Fro$ by sending each object to $\mathbb{R}$ and sending each morphism $c \to c'$ to the corresponding smooth function. We claim that $(X,C,F)$ is the colimit of this functor. The morphism $G(c) \to (X,C,F)$ is simply $c$ (note that $C = Hom_{Fro}(\mathbb{R},X)$ so $c$ is a morphism in $Fro$).

Now suppose that we have suitable morphisms $g_c \colon G(c) \to (Y,C_Y,F_Y)$. For each $x \in X$, there is a constant curve $c_x \colon \mathbb{R} \to X$ at $x$ (these are characterised by the fact that if $h \colon c \to c_x$ is a morphism in $\mathcal{C}$ then $c = c_x$). Consider $g_{c_x} \colon \mathbb{R} \to Y$. We shall show that this is a constant curve in $Y$. Let $h \in C^\infty(\mathbb{R},\mathbb{R})$ and examine $g_{c_x} \circ h$. As the $g_c$ are compatible, $g_{c_x} \circ h = g_{c_x \circ h}$. But as $c_x$ is constant, $c_x \circ h = c_x$ so $g_{c_x} \circ h = g_{c_x}$ and thus $g_{c_x}$ is constant.

Define $h \colon X \to Y$ by $h(x) = g_{c_x}(0)$. This is a set map, let us show that it lifts to Frölicher spaces. To do this, we look at $h \circ c$ for a smooth curve $c \in C$. Let $t \in \mathbb{R}$ and let $x = c(t)$ in $X$. Then $(h \circ c)(t) = h(x) = g_{c_x}(0)$. Let $f_t \colon \mathbb{R} \to \mathbb{R}$ be the constant function at $t$. Then $c \circ f_t = c_x$ and so $g_c \circ f_t = g_{c_x}$. Thus $g_{c_x}(0) = (g_c \circ f_t)(0) = g_c(t)$. Hence $h \circ c = g_c$. As $g_c \colon \mathbb{R} \to (Y,C_Y,F_Y)$ is a morphism in the category of Frölicher spaces with source $\mathbb{R}$, it is an element of $C_Y$. Hence $h$ takes smooth curves in $X$ to smooth curves in $Y$ and so is a morphism of Frölicher spaces.

This also establishes $(X,C,F)$ as the colimit since we have the factorisation $g_c = h \circ c$.