nLab opposite category

Idea

For a category $C$, its opposite category $C^{op}$ is the category obtained by formally reversing the direction of all its morphisms (while retaining their original composition law).

Categories generalize (are a horizontal categorification of) monoids, groups and algebras, and forming the opposite category corresponds to forming the opposite of a group, of a monoid, of an algebra.

Definition

In category theory

For a category $C$, the opposite category $C^{op}$ has the same objects as $C$, but a morphism $f : x \to y$ in $C^{op}$ is the same as a morphism $f : y \to x$ in $C$, and a composite of morphisms $g f$ in $C^{op}$ is defined to be the composite $f g$ in $C$.

More precisely, $C_{\mathrm{obj}}$ and $C_{\mathrm{mor}}$ are, respectively, the collections of objects and of morphisms of $C$, and if the structure maps of $C$ are

• source and target: $s_C,t_C : C_{\mathrm{mor}} \to C_{\mathrm{obj}}$

• identity-assignment: $i_C : C_{\mathrm{obj}} \to C_{\mathrm{mor}}$

• composition: $\circ_C : C_{\mathrm{mor}} \times_{C_{\mathrm{obj}}} C_{\mathrm{mor}} \to C_{\mathrm{mor}}$

then $C^{op}$ is the category with

• the same (isomorphic) collections of objects and morphisms

  $(C^{op})_{\mathrm{obj}} := C_{\mathrm{obj}}$

  $(C^{op})_{\mathrm{mor}} := C_{\mathrm{mor}}$

• the same identity-assigning map

  $i_{C^{op}} := i_C$

• switched source and target maps

  $s_{C^{op}} := t_C$

  $t_{C^{op}} := s_C$

• the same composition operation,

  $\circ_{C^{op}} := \circ_{C}$.

  or more precisely, the composition operation of $C^{op}$ is

  $$\circ_{C^{op}} : C_{\mathrm{mor}}^{op} {}_{s^{op}}\times_{t^{op}} C_{\mathrm{mor}}^{op} = C_{\mathrm{mor}} {}_{t} \times_{s} C_{\mathrm{mor}} \stackrel{\simeq}{\to} C_{\mathrm{mor}} {}_{s} \times_{t} C_{\mathrm{mor}} \stackrel{\circ}{\to} C_{\mathrm{mor}} \,,$$

  where the isomorphism in the middle is the unique one induced from the universality of the pullback.

Notice that hence the composition law does not change when passing to the opposite category. Only the interpretation of in which direction the arrows point does change. So forming the opposite category is a completely formal process. Nevertheless, due to the switch of source and target, the opposite category $C^{op}$ is usually far from being equivalent to $C$. See the examples below.

If $V$ is a monoidal category, then $V^{op}$ is equivalent to $(\Sigma V)^{co}$ where $\Sigma V$ is the delooping of $V$, i.e. $V$ viewed as a one-object bicategory and ${co}$ is the opposite on 2-cells

Opposite functors

Given categories $C$ and $D$, the opposite functor of a functor $F:C\to D$ is the functor $F^{op}:C^{op}\to D^{op}$ such that $F^{op}_{obj}=F_{obj}$ and $F^{op}_{mor}=F_{mor}$.

In the literature, $F^{op}$ is often confused with $F$. This is unfortunate, since (for example) natural transformations $F^{op}\to G^{op}$ (of functors $C^{op}\to D^{op}$) can be identified with natural transformations $G\to F$ (and not $F\to G$).

Opposite natural transformations

Given categories $C$ and $D$, functors $F,G:C\to D$, the opposite natural transformation of a natural transformation $t:F\to G$ is the natural transformation $t^{op}:G^{op}\to F^{op}$, induced by the same map $C_{obj}\to D_{mor}$ as $t$.

Again, in the literature $t^{op}$ is often confused with $t$.

The oppositization 1-functor

If we let $Cat$ denote the 1-category of strict small categories with functors between them, then there is an functor $op \colon Cat \to Cat$ sending each category to its opposite and each functor to its opposite, and

$$op^2 \;=\; id_{Cat} \,.$$

In fact, up to natural isomorphism, there are only two functors $F \colon Cat \to Cat$ that are equivalences: the identity functor $id_{CaT}$ and the oppositization functor $op$. In other words, the automorphism 2-group $Aut(Cat)$ of all autoequivalences of the 1-category $Cat$ is equivalent to the group $\mathbb{Z}/2$ viewed as a 0-truncated 2-group.

To see this, note that any autoequivalence of the 1-category $Cat$ fixes the terminal object $\ast$ up to unique isomorphism, and the arrow category 2 is the unique minimal generator (i.e. it is a generator and no proper subobject is a generator) so it is also fixed up to isomorphism. Since every category is functorially a colimit of copies of 2, once we know whether the autoequivalence fixes or swaps the two maps $1 \to 2$ the autoequivalence is determined up to a natural isomorphism, so every such autoequivalence is naturally isomorphic to either the identity or $op$.

For this argument and related questions touching on higher category theory see Toën (2005), Thm. 6.3; Barwick & Schommer-Pries (2011,21), Rem. 13.16; Ara, Groth & Gutiérrez (2013, 15) (cf. also Campion 2015).

The oppositization 2-functor

There are many advantages to treating $Cat$ as a 2-category with natural transformations as 2-morphisms. Then the above three constructions of the opposite category, opposite functor, and opposite natural transformation combine together to give the oppositization 2-functor

$$op: Cat^{co}\to Cat,$$

where $Cat^{co}$ denotes the 2-cell dual of the 2-category $Cat$, with the direction of 2-morphisms reversed and the direction of 1-morphisms preserved.

In enriched category theory

The definition has a direct generalization to enriched category theory.

For $V$ a symmetric monoidal category and $C$ a $V$-enriched category the opposite $V$-enriched category $C^{op}$ is defined to be the $V$-enriched category with the same objects as $C$ and with

$$C^{op}(c,d) := C(d,c)$$

and composition given by

$$C^{op}(b,c)\otimes C^{op}(a,b) := C(c,b) \otimes C(b,a) \stackrel{\sigma}{\to} C(b,a) \otimes C(c,b) \stackrel{\circ_C}{\to} C_{c,a} =: C^{op}(a,c) \,.$$

The unit maps $j_a : I \to C^{op}(a,a)$ are those of $C$ under the identification $C^{op}(a,a) = C(a,a)$.

Note that the braiding of $V$ is used in defining composition for $C^{op}$. So, we cannot define the opposite of a $V$-enriched category if $V$ is merely a monoidal category, though $V$-enriched categories still make perfect sense in this case. If $V$ is a braided monoidal category there are (at least) two ways to define “$C^{op}$”, resulting in two different “opposite categories”: we can use either the braiding or the inverse braiding. If $V$ is symmetric these two definitions coincide.

The opposite category can be regarded as a dual object of $C$ in the monoidal bicategory $V Prof$ of $V$-categories and $V$-profunctors. (Note that this does not characterize $C^{op}$ up to equivalence, but only up to Morita equivalence, i.e. up to Cauchy completion.) When $V$ is symmetric, then $V Prof$ is also symmetric monoidal, so there is only one notion of dual object. When $V$ is braided, then $V Prof$ is not symmetric and has two notions of dual: a left dual and a right dual. These are exactly the two different opposite categories referred to above (the “left opposite” and “right opposite”).

In higher category theory

See

• opposite 2-category

• opposite (∞,1)-category

The nerve of the opposite category

The nerve $N(C^{op})$ of $C^{op}$ is the simplicial set that is degreewise the same as $N(C)$, but in each degree with the order of the face and the order of the degeneracy maps reversed. See opposite quasi-category for more details.

Classes of examples

Opposite group

For $G = (S, \cdot)$ a group (or monoid or associative algebra, etc.) with product operation

$$\cdot_G : S \times S \to S$$

the opposite group $G^{op}$ is the group whose underlying set (underlying object, underlying vector space, etc.) is the same as that of $G$

$$S^{op} := S$$

but whose product operation is that of $G$ but combined with a switch of the order of the arguments:

$$\cdot_{G^{op}} : S \times S \stackrel{\sigma}{\to} S \times S \stackrel{\cdot_G}{\to} S \,.$$

So for $g,h \in S$ two elements we have that their product in the opposite group is

$$g \cdot_{G^{op}} h := h \cdot_{G} g \,.$$

Now, the group $G$ may be thought of as the pointed one-object delooping groupoid $\mathbf{B}G$ which is the groupoid with a single object, with $S$ as its set of morphisms, and with $\cdot_G$ its composition operation.

Under this identification of groups with one-object categories, passing to the opposite category corresponds precisely to passing to the opposite group

$$(\mathbf{B}G)^{op} = \mathbf{B}(G^{op}) \,.$$

Opposite of the opposite

The opposite of an opposite category is the original category:

$$(C^{op})^{op} = C \,.$$

This is also true for $V$-enriched categories when $V$ is symmetric monoidal, but not when $V$ is merely braided. However, in the latter case we can say $(C^{op1})^{op2} = C = (C^{op2})^{op1}$, i.e. the two different notions of “opposite category” are inverse to each other (as is always the case for left and right dualization operations in a non-symmetric monoidal (bi)category).

Co-algebraic structures

Every algebraic structure in a category, for instance the notion of monoid in a monoidal category $C$, has a co-version, where in the original definition the direction of all morphisms is reversed – for instance the co-version of a monoid is a comonoid. Of an algebra its a coalgebra, etc.

One may express this succinctly by saying that a co-structure in $C$ is an original structure in $C^{op}$. For instance a comonoid in $C$ is a monoid in $C^{op}$.

Duality

Passing to the opposite category is a realization of abstract duality.

This goes as far as defining some entities as objects in an opposite category–in particular, all generalizations of geometry which characterize spaces in terms of algebras. The idea of noncommutative geometry is essentially to define a category of spaces as the opposite category of a category of algebras. Similarly, a locale is opposite to a frame.

Are there examples where algebras are defined as dual to spaces?

Another example is the definition of the category of $L_\infty$-algebroids as that opposite to quasi-free differential graded algebras, identifying every $L_\infty$-algebra with its dual Chevalley-Eilenberg algebra.

Specific examples

Opposite of $Set$ and $FinSet$

The power set-functor

$$\mathcal{P} \;\colon\; Set^{op} \to Bool$$

constitutes an equivalence of categories from the opposite category of Set to that of complete atomic Boolean algebras. See at Set – Properties – Opposite category and Boolean algebras

Restricted to finite sets this says that the opposite of the category FinSet of finite sets is equivalent to the category of finite boolean algebras

$$FinSet^{op} \simeq FinBoolAlg \,.$$

See at FinSet – Properties – Opposite category. See at Stone duality for more.

Related concepts

• opposite 2-category

• opposite model category

• opposite (infinity,1)-category

• opposite adjunction

References

The basic notion – §II.2 of:

• Saunders MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

In the generality of enriched category theory – p. 12 of:

• Max Kelly, Basic Concepts of Enriched Category Theory, Lecture Notes in Mathematics 64, Cambridge University Press (1982)

  Republished as: Reprints in Theory and Applications of Categories 10 (2005) 1-136 [tac:10, pdf]

On classification of the autoequivalences of $Cat$ (and more generally of $(\infty,1)Cat$, $(\infinity,n)Cat$ and of $(\infty,1)Operad$):

• Bertrand Toën, Vers une axiomatisation de la théorie des catégories supérieures, K-Theory 34 3 (2005) 233-263 [arXiv:math/0409598, doi:10.1007/s10977-005-4556-6]

• Clark Barwick, Chris Schommer-Pries, Rem. 13.16 in: On the Unicity of the Homotopy Theory of Higher Categories, J. Amer. Math. Soc. 34 (2021) 1011-1058 [arXiv:1112.0040, slides, doi:10.1090/jams/972]

• Dimitri Ara, Moritz Groth, Javier J. Gutiérrez: On autoequivalences of the $(\infty,1)$-category of $\infty$-operads, Mathematische Zeitschrift 281 3 (2015) 807-848 [arXiv:1312.4994, doi:10.1007/s00209-015-1509-5]

and discussion for the case of Ho(Cat):

• Tim Campion, Does the category of categories-mod-natural-isomorphism have any nonobvious autoequivalences? (2015) [MO:q/223424]