nLab equivalence of categories

Idea

The concept of equivalence of categories is the correct category theoretic notion of “sameness” of categories.

Concretely, an equivalence between two categories is a pair of functors between them which are inverse to each other up to natural isomorphism of functors (inverse functors).

This is like an isomorphism, but weakened such as to accomodate for the fact that the correct ambient context for categories is not iself a 1-category, but is the 2-category Cat of all categories. Hence abstractly an equivalence of categories is just the special case of an equivalence in a 2-category specialized to Cat.

If some foundational fine print is taken care of, then a functor exhibits an equivalence of categories precisely if it is both essentially surjective and fully faithful. This is true in classical mathematics if the axiom of choice is assumed. It remains true non-classically, say for internal categories, if the concept of functor is suitably adapted (“anafunctors”), or the concept of essentially surjective is suitably adapted (“split essentially surjective”).

From the point of view of logic one may say that two categories are equivalent if they have the same properties — although this only applies (by definition) to properties that obey the principle of equivalence.

Just as equivalence of categories is the generalization of isomorphism of sets from sets to categories, so the concept generalizes further to higher categories (see e.g. equivalence of 2-categories, equivalence of (∞,1)-categories) and ultimately to equivalence of $\infty$-categories.

Definitions

Definition

An equivalence between two categories $\mathcal{C}$ and $\mathcal{D}$ is an equivalence in the 2-category Cat of all categories, hence a pair of inverse functors, hence it is

1. a pair of functors

   $$\mathcal{C} \underoverset {\underset{\phantom{AA}F \phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}G\phantom{AA}}{\longleftarrow}} {} \mathcal{D},$$

2. natural isomorphisms

   $$F \circ G \cong Id_{\mathcal{D}}$$

   and

   $$G \circ F \cong Id_{\mathcal{C}} \,.$$

This is called an adjoint equivalence if the natural isomorphisms above satisfy the triangle identities, thus exhibiting $F$ and $G$ as a pair of adjoint functors.

Two categories are called equivalent if there exists an equivalence between them.

Proposition

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a functor. Then the following are equivalent:

1. $F$ is part of an equivalence of categories in the sense of def.

2. $F$ is

   1. a split essentially surjective functor and

   2. a fully faithful functor.

Proof

This proof is from the HoTT book, and is set in the context of homotopy type theory. The categories here are what the HoTT book calls “precategories”, i.e. not univalent categories:

Suppose $F$ is an equivalence of categories with $G,\eta,\epsilon$ specified. Then we have the function

$$\begin{aligned} hom_B(F a, F b) &\to hom_A(a,b), \\ g &\mapsto \eta_b^{-1}\circ G(g) \circ \eta_a. \end{aligned}$$

For $f:hom_A(a,b)$, we have

$$\eta_b^{-1} \circ G(F(f)) \circ \eta_a = \eta_b^{-1} \circ \eta_b \circ f = f$$

while for $g: hom_B(F a, F b)$ we have

$$\begin{aligned} F(\eta_b^{-1} \circ G(g) \circ \eta_a) &= F(\eta_b^{-1}) \circ F(G(g)) \circ F(\eta_a) \\ &= \epsilon_{F b} \circ F(G(g)) \circ F(\eta_a) \\ &= g \circ \epsilon_{F a} \circ F(\eta_a) \\ &= g \end{aligned}$$

using naturality of $\epsilon$, and the triangle identities twice. Thus, $F_{a,b}$ is an equivalence, so $F$ is fully faithful. Finally, for any $b:B$, we have $G b : A$ and $\epsilon_b : F G b \cong b$.

On the other hand, suppose $F$ is fully faithful and split essentially surjective. Define $G_0:B_0\to A_0$ by sending $b:B$ to the $a:A$ given by the specified essential splitting, and write $\epsilon_b$ for the likewise specified isomorphism $F G b \cong b$.

Now for any $g: hom_B(b,b')$, define $G(g): hom_A(G b, G b')$ to be the unique morphism such that $F(G(g))=(\epsilon_{b'})^{-1} \circ g \circ \epsilon_b$ which exists since $F$ is fully faithful. Finally for $a:A$ define $\eta_a : hom_A(a,G F a)$ to be the unique morphism such that $F \eta_a = \epsilon^{-1}_{F a}$. It is easy to verify that $G$ is a functor and that $(G,\eta \epsilon)$ exhibit $F$ as an equivalence of categories.

We clearly recover the same function $G_0 : B_0 \to A_0$. For the action of $F$ on hom-sets, we must show that for $g:hom_B (b,b')$, $G(g)$ is the necessarily unique morphism such taht $F(G(g))=(\epsilon_{b'})^{-1} \circ g \circ \epsilon_b$.

But this holds by naturality of $\epsilon$. Then we show $(2) \to (1) \to (2)$ gives the same data hence $(1)\simeq (2)$.

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 $Cat$ as a mere 1-category; it is too strong and is really only of interest for strict categories. The next, strong equivalence, comes from viewing $Cat$ 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 $Cat$ as a model category; it is correct with or without choice and is about as simple to define as strong equivalence. The fourth, anaequivalence, comes from viewing $Cat$ 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 is 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 $F\colon C \to D$ and $G\colon D \to C$ such that $F G$ and $G F$ 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 $Cat$ 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. For example, here are some basic and important equivalences of categories that are not isomorphisms:

• Let $C$ be the category of pointed sets, and let $D$ be the category of sets and partial functions. The functor $F:C\to D$ takes a pointed set to its subset of non-basepoint elements, and a pointed function to the induced partial function on these (which is defined on those elements not sent to the basepoint). See the section “The category of sets and partial functions” in partial function for this equivalence.

• Let $C$ be the category of finite-dimensional vector spaces over a field $k$, and let $D$ be the category whose objects are natural numbers and whose morphisms $n\to m$ are $m\times n$ matrices of elements of $k$ (which is equivalently the full subcategory of $C$ spanned by the specific vector spaces $k^n$). Note in particular that here $D$ is a small category, while $C$ is not (though it is essentially small, being equivalent to $D$).

Strong equivalence

Two strict categories $C$ and $D$ are strongly equivalent if there exist strict functors $F\colon C \to D$ and $G\colon D \to C$ such that $F G$ and $G F$ 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 $Cat$ 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.

If possible, use or modify the counterexample to isomorphism to show how choice follows if strong equivalence is assumed correct.

Weak equivalence

Two categories $C$ and $D$ are weakly equivalent if there exist a category $X$ and functors $F\colon X \to D$ and $G\colon X \to C$ 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 a weak equivalence from $X$ to $C$) and call the span $(X,F,G)$ a weak equivalence between $C$ and $D$. (It is not entirely trivial to check that such spans can be composed, but they can be.)

A strict 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 $Cat$ 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 $F\colon C \to D$ and $G\colon D \to C$ such that $F G$ and $G F$ 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 $Cat$ 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 $Cat$ above as obtained from the $2$-category $Str Cat$ of strict categories, strict functors, and natural transformations by formally inverting the weak equivalences as in homotopy theory.

Fully faithful essentially surjective functors

Finally, there are fully faithful and essentially surjective functors. However, while in general, these are not the same as equivalences in all mathematical foundations, they are the same under certain restrictions:

Proposition

Assume the ambient context is one of the following:

• classical mathematics with the axiom of choice;

• constructive or internal category theory with “functor” meaning anafunctor;

• higher-level foundations with “category” meaning univalent category.

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a functor. Then the following are equivalent:

1. $F$ is part of an equivalence of categories in the sense of def.

2. $F$ is

   1. an essentially surjective functor and

   2. a fully faithful functor.

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 homotopy type theory, some other forms of 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 $Cat$, 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 $F G \cong 1$ and $1 \cong G F$ 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 $F G \cong 1$ and $1 \cong G F$)

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.

An example of a non-adjoint equivalence

Identify a group $H$ with its delooping. One can check the following:

• Any equivalence $F : H \leftrightarrows H : G$ of a group with itself comprises two automorphisms $F, G$, such that $F G$ and $G F$ are inner. The unit and counit are the group elements $g_{\rho}$ such that $GF(k) = g_{\rho} k g_{\rho}^{-1}$ and $g_{\sigma}$ such that $FG(k) = g_{\sigma}^{-1} k g_{\sigma}$ for any $k \in H$.

• Any equivalence of $H$ with itself where $F$ and $G$ are themselves also inner is an adjoint equivalence.

• If $H$ has trivial center, then any equivalence of $H$ with itself is an adjoint equivalence.

• To obtain a non-adjoint equivalence, we therefore need a group $H$ with nontrivial center and nontrivial outer automorphisms, such that we can pick two whose products are inner.

• So take $H = K$ the Klein 4-group. This is a product of abelian groups, so abelian, so is its own center. In fact, it’s $\mathbb{Z}/2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z}$, so let $F = G$ the automorphism which interchanges coordinates. $FG = GF = \operatorname{id}_{K}$, which is given by conjugation by any element.

• If this were adjoint, the triangle equality for $F$ will stipulate that $F(g_{\rho}) = g_{\sigma}^{-1}$. We can pick $g_{\rho}$ and $g_{\sigma}$ to break this. For example, let $g_{\rho}$ be $(1,1)$ and let $g_{\sigma}$ be $(0,1)$.

This is a special case of the fact that, given a non-adjoint equivalence, you can always replace its unit with another unit (which determines the counit) to improve the equivalence to an adjoint equivalence.

In higher categories

All of the above types of equivalence make sense for $n$-categories and $\infty$-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 $Bicat$ 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 even if it is, the equivalence transformations need not be strictly $2$-natural. Thus, it need not be an equivalence in the strict 3-category $Str 2 Cat$ of $2$-categories, strict $2$-functors, and strict $2$-natural transformations, or even in the semi-strict 3-category? $Gray$ of strict $2$-categories, strict $2$-functors, and pseudonatural transformations.

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

Related concepts

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

• 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