## 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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