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Any number of constructions guarantee the existence of maps $f$ without guaranteeing their uniqueness. Some time ago, I was introduced to the terminology "versal" for such a construction. I wonder: …

If $M$ is a monoid object in a pointed category $\mathcal{C}$, then a right $M$-module is an object $X$ equipped with a morphism $\alpha: X\times M\to X$ that satisfies the usual rules. There are als …

This has to do with the "pushout-product" construction. In a category $\mathcal{C}$, suppose we have $C\gets A\to B$ with pushout $D$ and $Y\gets W\to X$ with pushout $Z$. Then we can form $$ (C\tim …

A monoid object in a pointed category $\mathcal{C}$ is an object $M$ equipped with a multiplication morphism $\mu: M\times M\to M$ that is associative and unital, meaning that the diagrams that expres …

No. For example, if $X$ is a CW complex with skeleta $X_n$ and $f_n\simeq*$, then $f$ is a phantom map. Their homotopy classes are in bijective correspondence with $\lim^1 [\Sigma X_n, Y]$, and ar …

I think this doesn't quite work: Let $\mathcal{C}$ be the category whose objects are the point of $X$, and define $$ \mathrm{mor}_\mathcal{C}(x,y) = \{ \mbox{closed sets containing both $x$ and $y$} …

Group actions! Treat $G$ as a category $\mathcal{G}$. Then a $G$-action is a functor $\mathcal{G}\to \mathcal{C}$. Its colimit is the orbit space and its limit is the fixed points.

Introductory algebra courses tend to systematically confuse products with coproducts, and more generally, confuse targets with domains. This systematically causes confusion in students (what is the d …

The join of $A$ and $B$ is the pushout of the diagram $$ CA \times B \gets A\times B \to A\times CB, $$ which can be formulated in either the pointed or unpointed topological category. This pushout is …

I've written a paper (or two) about collection $\mathcal{R}$ of all pointed topological spaces $Y$ satisfying the property $\mathrm{map}_*(X,Y) \sim *$ (for fixed $X$). The interesting fact is that …