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

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

Now suppose that we have suitable morphisms g c:G(c)(Y,C Y,F Y)g_c \colon G(c) \to (Y,C_Y,F_Y). For each xXx \in X, there is a constant curve c x:Xc_x \colon \mathbb{R} \to X at xx (these are characterised by the fact that if h:cc xh \colon c \to c_x is a morphism in 𝒞\mathcal{C} then c=c xc = c_x). Consider g c x:Yg_{c_x} \colon \mathbb{R} \to Y. We shall show that this is a constant curve in YY. Let hC (,)h \in C^\infty(\mathbb{R},\mathbb{R}) and examine g c xhg_{c_x} \circ h. As the g cg_c are compatible, g c xh=g c xhg_{c_x} \circ h = g_{c_x \circ h}. But as c xc_x is constant, c xh=c xc_x \circ h = c_x so g c xh=g c xg_{c_x} \circ h = g_{c_x} and thus g c xg_{c_x} is constant.

Define h:XYh \colon X \to Y by h(x)=g c x(0)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 hch \circ c for a smooth curve cCc \in C. Let tt \in \mathbb{R} and let x=c(t)x = c(t) in XX. Then (hc)(t)=h(x)=g c x(0)(h \circ c)(t) = h(x) = g_{c_x}(0). Let f t:f_t \colon \mathbb{R} \to \mathbb{R} be the constant function at tt. Then cf t=c xc \circ f_t = c_x and so g cf t=g c xg_c \circ f_t = g_{c_x}. Thus g c x(0)=(g cf t)(0)=g c(t)g_{c_x}(0) = (g_c \circ f_t)(0) = g_c(t). Hence hc=g ch \circ c = g_c. As g c:(Y,C Y,F Y)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 YC_Y. Hence hh takes smooth curves in XX to smooth curves in YY and so is a morphism of Frölicher spaces.

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

Revised on December 17, 2010 09:39:45 by Andrew Stacey (129.241.15.133)