Equivalences, localizations, and inverting morphisms

A functor $F:\mathcal C\to\mathcal D$ is an equivalence when there is a functor $G$ in the other direction with natural isomorphisms $GF\cong\mathrm{Id}_{\mathcal C}$ and $FG\cong\mathrm{Id}_{\mathcal D}$. An isomorphism of categories would require strict inverse equalities instead. Equivalent categories can therefore differ by redundant isomorphic objects, while an isomorphism of categories gives a bijection on the actual object sets.

A full and faithful functor that is essentially surjective is an equivalence, with the usual choice of representatives and comparison isomorphisms; for small categories ordinary set-theoretic choice suffices. Essential surjectivity means every target object is isomorphic to an image object, rather than literally equal to one. Full faithfulness supplies the unique lifted arrows needed to build a quasi-inverse after representatives are chosen. Merely being surjective on objects would not recover missing arrows or distinguish identified arrows.

Let $\mathcal C$ be small and W a set of its arrows. A localization $Q:\mathcal C\to\mathcal C[W^{-1}]$ sends every arrow in W to an isomorphism. For each target category D, precomposition with Q is fully faithful, and its essential image consists of the W-inverting functors. Equivalently, it is an equivalence onto the full subcategory of those functors. The statement includes natural transformations as well as the existence of factorizations, not just an object-level assignment of functors.

One construction uses finite composable zigzags, permitting the selected arrows to be traversed backward, and quotients by composition, identity, and inverse relations. Smallness makes the relevant paths and their quotient into sets. In this explicit presentation, extensions can be made strictly unique by evaluating backward arrows as inverses. Arbitrary localizations express that uniqueness up to the appropriate natural isomorphism. Reducing every zigzag to a single fraction-shaped roof requires extra calculus-of-fractions hypotheses and is not automatic for an arbitrary W.

The one-object category with endomorphism monoid $\mathbb N$ under addition is generated by one arrow. Inverting that generator gives the one-object category with group $\mathbb Z$: a target functor is determined by an invertible endomorphism a, and the extension sends an integer n to its positive or negative power. No finite-order relation is imposed unless it already follows from the original relations. This is why formal inversion produces integers rather than residues modulo a chosen modulus.

A homomorphism from the integers to a finite cyclic group is a useful W-inverting representation, but it identifies distinct integers and cannot replace the universal construction. Similarly, making some arrows invertible can identify other arrows through the imposed relations; localization need not be faithful. Its guarantee concerns all W-inverting functors out of the original category. This mapping property determines the resulting category up to equivalence compatible with Q, without promising that its Hom sets retain their old sizes.