Equivalence of categories

Idea

Two categories are equivalent if they have the same properties —although this only applies (by definition) to properties that obey the principle of equivalence. An equivalence between categories is a functor that realises this.

If one works with internal categories or doubts the axiom of choice, some care must be taken in the definition.

Equivalence of categories generalises to higher categories and ultimately to equivalence of $\mathrm{\infty }$-categories.

Definitions

This is called an adjoint equivalence if in addition $F$ and $G$ form a pair of adjoint functors.

Variants

We discuss some possible variants of the definition of equivalence of categories, each of which comes naturally from a different view of Cat.

The first, isomorphism, comes from viewing $\mathrm{Cat}$ as a mere category; it is too strong and is really only of historical interest. The next, strong equivalence, comes from viewing $\mathrm{Cat}$ as a strict 2-category; it is the most common definition given and is correct if and only if the axiom of choice holds. The next definition, weak equivalence, comes from viewing $\mathrm{Cat}$ as a model category; it is correct with or without choice and is about as simple to define as strong equivalence. The last, anaequivalence, comes from viewing $\mathrm{Cat}$ as a bicategory that is not (without the axiom of choice) equivalent (as a bicategory!) to the strict $2$-category that defines strong equivalence; it is also always correct.

It's also possible to define ‘category’ in such a way that only a correct definition can be stated, but here we use the usual algebraic definitions of category, functor, and natural isomorphism.

Isomorphism

Two strict categories $C$ and $D$ are isomorphic if there exist strict functors $F:C\to D$ and $G:D\to C$ such that $FG$ and $GF$ are each equal to the appropriate identity functor. In this case, we say that $F$ is an isomorphism from $C$ to $D$ (so $G$ is an isomorphism from $D$ to $C$) and call the pair $(F,G)$ an isomorphism between $C$ and $D$. The functor $G$ is called the strict inverse of $F$ (so $F$ is the strict inverse of $G$).

If you think of $\mathrm{Cat}$ as the category of (strict) categories and functors, then this is the usual notion of isomorphism in a category. This is the most obvious notion of equivalence of categories and the first to be considered, but it is simply too strong for the purposes to which category theory is put.

Strong equivalence

Two strict categories $C$ and $D$ are strongly equivalent if there exist strict functors $F:C\to D$ and $G:D\to C$ such that $FG$ and $GF$ are each naturally isomorphic (isomorphic in the relevant functor category) to the appropriate identity functor. In this case, we say that $F$ is a strong equivalence from $C$ to $D$ (so $G$ is a strong equivalence from $D$ to $C$). The functor $G$ is called a weak inverse of $F$ (so $F$ is a weak inverse of $G$).

Note that an isomorphism is precisely a strong equivalence in which the natural isomorphisms are identity natural transformations.

If you think of $\mathrm{Cat}$ as the strict 2-category of (strict) categories, functors, and natural transformations, then this is the usual notion of equivalence in a $2$-category. This is the first ‘correct’ definition of equivalence to be considered and the one most often seen today; it is only correct using the axiom of choice.

Weak equivalence

Two strict categories $C$ and $D$ are weakly equivalent if there exist a category $X$ and strict functors $F:X\to D$ and $G:X\to C$ that are essentially surjective and fully faithful. In this case, we say that $F$ is a weak equivalence from $X$ to $D$ (so $G$ is an equivalence from $X$ to $C$) and call the span $(X,F,G)$ a weak equivalence between $C$ and $D$.

A functor with a weak inverse is necessarily essentially surjective and fully faithful; the converse is equivalent to the axiom of choice. Thus any strong equivalence becomes a weak equivalence in which $X$ is taken to be either $C$ or $D$ (or even built symmetrically out of $C$ and $D$ if you're so inclined); a weak equivalence becomes a strong equivalence using the axiom of choice to find weak inverses and composing across $X$.

If you think of $\mathrm{Cat}$ as the model category of categories and functors with the canonical model structure, then this is the usual notion of weak equivalence in a model category.

Anaequivalence

Two categories $C$ and $D$ are anaequivalent if there exist anafunctors $F:C\to D$ and $G:D\to C$ such that $FG$ and $GF$ are each ananaturally isomorphic (isomorphic in the relevant anafunctor category) to the appropriate identity anafunctor. In this case, we say that $F$ is an anaequivalence from $C$ to $D$ (so $G$ is an anaequivalence from $D$ to $C$). The functor $G$ is called an anainverse of $F$ (so $F$ is an anainverse of $G$). See also weak equivalence of internal categories.

Any strict functor is an anafunctor, so any strong equivalence is an anaequivalence. Using the axiom of choice, any anafunctor is ananaturally isomorphic to a strict functor, so any anaequivalence defines a strong equivalence. Using the definition of an anafunctor as an appropriate span of strict functors, a weak equivalence defines two anafunctors which form an anaequivalence; conversely, either anafunctor in an anaequivalence is (as a span) a weak equivalence.

If you think of $\mathrm{Cat}$ as the bicategory of categories, anafunctors, and ananatural transformations, then this is the usual notion of equivalence in a $2$-category. It's fairly straightforward to turn any discussion of functors and strong equivalences in a context where the axiom of choice is assumed into a discussion of anafunctors and anaequivalences in a more general context.

We can also regard the $2$-category $\mathrm{Cat}$ above as obtained from the $2$-category $\mathrm{Str}\mathrm{Cat}$ of categories, functors, and natural transformations using homotopy theory by “formally inverting” the weak equivalences.

Remarks

Note that weak inverses go with strong equivalences. The terminology isn't entirely inconsistent (weak inverses contrast with strict ones, while weak equivalences contrast with strong ones) but developed at different times. The prefix ‘ana‑’ developed last and is perfectly consistent.

If you accept the axiom of choice, then you don't have to worry about the different kinds of equivalence (as long as you don't use isomorphism). This is not just a question of foundations, however, since the axiom of choice usually fails in internal contexts.

It's also possible to use foundations (such as type theory or FOLDS) in which isomorphism and strong equivalence are impossible to state. In such a case, one usually drops the prefixes ‘weak’ and ‘ana‑’. In the $n$-Lab, we prefer to remain agnostic about foundations but usually drop these prefixes as well, leaving it up to the reader to insert them if necessary.

Adjoint equivalence

Any equivalence can be improved to an adjoint equivalence: a strong equivalence or anaequivalence whose natural isomorphisms satisfy the triangle identities. In that case, $G$ is called the adjoint inverse of $F$ (so $F$ is the adjoint inverse of $G$). When working in the $2$-category $\mathrm{Cat}$, a good rule of thumb is that it is okay to consider either

• a functor with the property of being a general equivalence or
• a functor with the structure of being an adjoint equivalence (that is, a functor $G$ and a pair of natural isomorphisms $FG\cong 1$ and $1\cong GF$ satisfying the triangle identities),

whereas considering

• a functor with the structure of being a general equivalence (that is, merely a functor $G$ and a pair of natural isomorphisms $FG\cong 1$ and $1\cong GF$)

is fraught with peril. For instance, an adjoint inverse is unique up to unique isomorphism (much as a strict inverse is unique up to equality), while a weak inverse or anainverse is not. Including adjoint equivalences is also the only way to make a higher-categorical structure completely algebraic.

In higher categories

All of the above types of equivalence make sense for $n$-categories and $\mathrm{\infty }$-categories defined using an algebraic definition of higher category; again, it is the weak notion that is usually wanted. When using a geometric definition of higher category, often isomorphism cannot even be written down, so equivalence comes more naturally.

In particular, one expects (although a proof depends on the exact definition and hasn't always been done) that in any $(n+1)$-category of $n$-categories, every equivalence (in the sense of an $(n+1)$-category) will be essentially $k$-surjective for all $0\le k\le n+1$; this is the $n$-version of “full, faithful, and essentially surjective.” The converse should be true assuming that

• we have an axiom of choice and use weak (pseudo) $n$-functors, or
• we use $n$-anafunctors? (which are automatically weak).

If we use too strict a notion of $n$-functor, then we will not get the correct notion of equivalence; if we use weak $n$-functors but not anafunctors, then we will get the correct notion of equivalence only if the axiom of choice holds, although again this can be corrected by moving to a span. Note that even strict $n$-categories need weak $n$-functors to get the correct notion of equivalence between them!

For example, assuming choice, a strict 2-functor between strict $2$-categories is an equivalence in $\mathrm{Bicat}$ if and only if it is essentially (up to equivalence) surjective on objects, locally essentially surjective, and locally fully faithful. However, its weak inverse may not be a strict $2$-functor and the equivalence transformations need not be strictly 2-natural. Thus, it need not be an equivalence in the strict 3-category $\mathrm{Str}2\mathrm{Cat}$ of 2-categories, strict 2-functors, and strict 2-natural transformations, or even in the semi-strict 3-category? $\mathrm{Gray}$ of strict 2-categories, strict 2-functors, and pseudonatural transformations.

As with $\mathrm{Cat}$, we can recover $\mathrm{Bicat}$ as a full subtricategory of $\mathrm{Gray}$ by formally inverting all such weak equivalences. Note that even with the axiom of choice, $\mathrm{Bicat}$ is not equivalent (as a tricategory) to $\mathrm{Gray}$, even though by the coherence theorem for tricategories it is equivalent to some Gray-category; see here.

Related concepts

• equality, isomorphism, equivalence

• weak equivalence, homotopy equivalence, weak homotopy equivalence

• homotopy equivalence of toposes

• equivalence in an (∞,1)-category

• equivalence of categories, weak equivalence of internal categories

  adjoint functor

• equivalence of 2-categories, 2-adjunction

• equivalence of (∞,1)-categories, adjoint (∞,1)-functor