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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.

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

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