Even More Category Theory: The Elementary Topos

Theories and Theorems

In More Category Theory: The Grothendieck Topos, we defined the Grothendieck topos as something like a generalization of the concept of sheaves on a topological space. In this post we generalize it even further into a concept so far-reaching it can even be used as a foundation for mathematics.

I. Definition of the Elementary Topos

We start by discussing the generalization of the universal constructions we defined in More Category Theory: The Grothendieck Topos, called limits and colimits.

Given categories $latex mathbf{J}$ and $latex mathbf{C}$, we refer to a functor $latex F: mathbf{J}rightarrow mathbf{C}$ as a diagram in $latex mathbf{C}$ of type $latex mathbf{J}$, and we refer to $latex mathbf{J}$ as an indexing category. We write the functor category of all diagrams in $latex mathbf{C}$ of type $latex mathbf{J}$ as $latex mathbf{C^{J}}$.

Given a diagram $latex F: mathbf{J}rightarrow mathbf{C}$, a cone to $latex F$ is an object $latex N$ of $latex mathbf{C}$…

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