Unity of Opposites

I have been thinking about the nerve-realization theorem. See here: https://ncatlab.org/nlab/show/nerve+and+realization This blog post is about my attempts to think through it. The theorem is just about the usual coend formula for the left Kan extension, except that it studies a complicated case where some of the categories are enriched and some are not; on the other hand it is simpler in that one of the functors is fixed to be some variant of the Yoneda embedding.  In their notation, $\mathcal{C}$ is enriched but $S$ may not be. For example, a simplicial Abelian group or a simplicial topological space are functors from an unenriched category to a category that is enriched ($\mathbf{Ab}$) or at least somewhat enriched (hom sets in $\mathbf{Top}$ can be equipped with the compact open topology) Let $M$ be a small category. Let $\mathcal{A}$ be a co-complete category. Let $T: M\to \mathcal{A}$ be a functor. Theorem:   Let $R : \mathcal{A}\to \hat{M}=[M^{op},\mathbf{Sets}]$ be def...

Topos theory leads with the slogan - ``unification of logic and geometry.'' This theory has been more effective in marketing than in actually giving effective cross-fertilization between the two fields. A subject in between $A$ and $B$ may draw on both fruitfully and give back to both fruitfully, but it is most exciting when it really serves as a bridge between $A$ and $B$, allowing you to transfer ideas back and forth. The notion of ``Kripke model'' is one of the oldest proposed semantics for forcing. Some category theorists have noticed that a Kripke model (let's say a model of a first order theory) over an index set $I$ with partial-ordering relation $\leq$ can be viewed as a functor from $I$ as a poset category into the category of sets, or a presheaf depending on your orientation of $I$. This observation has led to a development of a general forcing semantics in an elementary topos; a good reference is Sheaves in Geometry and Logic by Mac Lane and Moerdijk, in ...