I have never really done any category theory before, and am looking to use it to somehow classify some of my work. In particular I am trying to determine wheter two given categories, say $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are equivalent or not. Are there any efficient ways of doing this?
Obviously if I define a functor $F:\mathcal{C}_{1}\rightarrow\mathcal{C}_{2}$, then I can check whether this is an equivalence of categories (either by using the definition, or by checking whether $F$ is full, faithful and every element of $\mathcal{C}_{2}$ is the isomorphic image of an element of $\mathcal{C}_{1}$), but this relies on defining the functor $F$. Are there any methods of determining whether two categories are equivalent/inequivalent without defining any functors?
Edit/Motivation: The case that I am dealing with involves trying to see if given two simplicial complexes $\Delta_{1}\subseteq\Delta_{2}$ with an associated $G$-action for some group $G$, the category of presheaves defined on $\Delta_{1}$ and the category of presheaves defined on $\Delta_{2}$ are equivalent (see Ronan and Smith's paper for details about presheaves). The motivation stems from the fact that there is a notion of simplicial complexes being $G$-homotopy equivalent, and I am trying to see whether I can generalise this to presheaves.