I have to find in terms of standard spaces (that is interval, circle, disk, sphere, cone, etc.) the reduced cone (i.e. cone in category of pointed spaces) $\operatorname{Cone}(*)$ where $\{*\}$ is one point set.
We aren't discussing any topology.
Sorry for my English, it's not my native language.
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$\begingroup$The reduced cone over a pointed space is (as I recall) the cone over the space with the fiber over the base point collapsed. Precisely, if $(X,x_0)$ is a topological space, its reduced cone is defined by $$(X\times I)/(X\times\{0\}\cup \{x_0\}\times I).$$ At this stage, it will probably be easier for you to draw a picture for yourself and figure out the answer. I'll leave you with a tip: In practice, the way I think about the reduced cone is by first taking the cone, then doing an additional collapse along the fiber over the base point. So take the cone of $\{*\}$ and then collapse out the fiber over the base point of that space.
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