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  • Diagrams, Categorical Limits, and Topoi (Part 1)

    Definition. A (small)¬†directed graph [latex]G[/latex] is a 3-tuple [latex](V,E,d)[/latex] where [latex]V[/latex] and [latex]E[/latex] are sets and [latex]d:E\to V\times V[/latex]. Similarly, a subgraph [latex]A[/latex] of [latex]G[/latex] is a 3-tuple [latex](V_A,E_A,d)[/latex] such that [latex]E_A[/latex] is contained in [latex]E[/latex] and [latex]V_A\times V_A\subseteq V\times V[/latex] contains [latex]d(E_A)[/latex]. Definition. If [latex]V\times V[/latex] is replaced with [latex]V\times V / \{(x,y)\sim (y,x)\}[/latex],…