# Tag Archives: homotopy

## A is Homotopy Equivalent to A^op via Functors

Let $\mathcal{A}$ be a small category and $\mathbf{B}\mathcal{A}$ its geometric realisation. It is evident that $\mathbf{B}\mathcal{A}$ and $\mathbf{B}\mathcal{A}^\circ$ are homotopy equivalent, and in fact homeomorphic. However, can we find functors that realise this equivalence? This post summarises some informal notes I have written on this following D. Quillen's paper Higher Algebraic […]

## Homotopy on Chain Complexes

Let $R$ be any associative ring with unit and $A$ an $R$-module. If $P$ is a projective module and $A\to P\to A = 1_A$, is $A$ necessarily projective?