# What is notion of diagram and cone from category theory to topological space

I was reading Steve Awodey's book Category Theory and this post.

In the book Category Theory, the author Steve Awodey introduced diagram on the page $$101:$$

Let $$\textbf{J}$$ and $$\textbf{C}$$ be categories. A diagram of type $$\textbf{J}$$ in $$\textbf{C}$$ is a functor. $$D:\textbf{J} \to \textbf{C}.$$ We write the objects in the "index category" $$\textbf{J}$$ lower case, $$i,j,...$$ and the values of the functor $$D: \textbf{J} \to \textbf{C}$$ in the form $$D_i,D_j,$$ etc.

A cone to a diagram $$D$$ consists of an object $$C$$ in $$\textbf{C}$$ and a family of arrows in $$\text{C},$$ $$c_j:C \to D_j$$ one for each object $$j \in J,$$ such that for each arrow $$\alpha:i \to j$$ in $$\textbf{J},$$ the following triangle commutes: $$\begin{array}{ccc} &C & \\ \swarrow^{c_i}&&\searrow^{c_j}\\ D(i)&\rightarrow^{D_{\alpha}}&D(j) \end{array}$$

I really have no idea of designing commutative diagram in latex. But the key thing is whether there is a notion of diagram and cone in topological space. I am curious because in the later section, Awodey discussed the continuity of functors. And I am interested in the diagram and cone because I think there might be some kind of concept about them in topological space.

• I don't understand. Let $\mathbf{C}$ be the category of topological spaces and apply the definitions – Randall Feb 13 at 2:21
• @Randall Then what are the diagrams then? – Zack Ni Feb 13 at 2:22
• Spaces and continuous maps? Are you asking if there is a notion of cone in spaces that existed "before" the categorical definition? – Randall Feb 13 at 2:25
• @Randall Yes, that is exactly what I am asking for. – Zack Ni Feb 13 at 2:26
• Also, continuity of functors will just mean preservation of limits. It won't abstract continuity from topology. – Randall Feb 13 at 2:30