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Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The …

Topos theory can be seen as a categorification of topology via the following analogies. \begin{array}{|c|c|} \hline \text{locales}&\text{Grothendieck toposes}\\\hline \text{open sets}&\text{sheaves}\ …

Given a topology on a set $X$, let $2^X$ be the poset of subsets of $X$ ordered by inclusion. Then the interior operator for the topology is a comonad on $2^X$. In fact the topologies on $X$ correspon …

Given a model of set theory $V$ there are various ways to construct a model in which the Axiom of Choice holds, such as Gödel's constructible universe $L^V$ or by using forcing*. I'm wondering if any …

Benjamin Steinberg's answer covers the case of topological spaces, so I'll answer this question for sites. The functor $$\mathrm{Sh}:\mathbf{Site}\rightarrow\mathbf{Topos}^{\mathrm{op}}$$ (here $\mat …

This is a bit of a boring answer, but such a functor always exists. We can use the functor that sends $0$ to $a$, every thing else in $[0,1]$ to $b$ and then sends the morphisms to $\mathrm{id}_a$, $\ …

Clearly either of $$\overset{X}{\longrightarrow}\fbox{$A$}\overset{Y}{\longrightarrow}\\+\\ \overset{X}{\longrightarrow}\fbox{$B$}\overset{Y}{\longrightarrow}$$ or $$\overset{X}{\longrightarrow}\fb …

The issue here is that the inverses to $V\rightarrow V^{**}$ and $V^*\otimes V\rightarrow \mathrm{Hom}(V,V)$ don't exist in the infinite dimensional case. So in order to show that they exist one has t …