nLab
opposite category

Contents

Idea

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

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

  • source and target: s C,t C:C morC objs_C,t_C : C_{\mathrm{mor}} \to C_{\mathrm{obj}}

  • identity-assignment: i C:C objC mori_C : C_{\mathrm{obj}} \to C_{\mathrm{mor}}

  • composition: C:C mor× C objC morC mor\circ_C : C_{\mathrm{mor}} \times_{C_{\mathrm{obj}}} C_{\mathrm{mor}} \to C_{\mathrm{mor}}

then C opC^{op} is the category with

  • the same (isomorphic) collections of objects and morphisms

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

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

  • the same identity-assigning map

    i C op:=i Ci_{C^{op}} := i_C

  • switched source and target maps

    s C op:=t Cs_{C^{op}} := t_C

    t C op:=s Ct_{C^{op}} := s_C

  • the same composition operation,

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

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

    C op:C mor op s op× t opC mor op=C mor t× sC morC mor s× tC morC mor, \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 opC^{op} is usually far from being equivalent to CC. See the examples below.

In enriched category theory

The definition has a direct generalization to enriched category theory.

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

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

and composition given by

C op(b,c)C op(a,b):=C(c,b)C(b,a)σC(b,a)C(c,b) CC c,a=:C op(a,c). 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:IC op(a,a)j_a : I \to C^{op}(a,a) are those of CC under the identification C op(a,a)=C(a,a)C^{op}(a,a) = C(a,a).

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

The opposite category can be regarded as a dual object of CC in the monoidal bicategory VProfV Prof of VV-categories and VV-profunctors. (Note that this does not characterize C opC^{op} up to equivalence, but only up to Morita equivalence, i.e. up to Cauchy completion.) When VV is symmetric, then VProfV Prof is also symmetric monoidal, so there is only one notion of dual object. When VV is braided, then VProfV 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

Properties

The nerve N(C op)N(C^{op}) of C opC^{op} is the simplicial set that is degreewise tthe same as N(C)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,)G = (S, \cdot) a group (or monoid or associative algebra, etc.) with product operation

G:S×SS \cdot_G : S \times S \to S

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

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

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

G op:S×SσS×S SS. \cdot_{G^{op}} : S \times S \stackrel{\sigma}{\to} S \times S \stackrel{\cdot_S}{\to} S \,.

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

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

Now, the group GG may be thought of as the pointed one-object delooping groupoid BG\mathbf{B}G which is the groupoid with a single object, with SS as its set of morphisms, and with G\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

(BG) op=B(G op). (\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. (C^{op})^{op} = C \,.

This is also true for VV-enriched categories when VV is symmetric monoidal, but not when VV is merely braided. However, in the latter case we can say (C op1) op2=C=(C op2) op1(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 CC, 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 CC is an original structure in C opC^{op}. For instance a comonoid in CC is a monoid in C opC^{op}.

Duality

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

This goes as far that some entities are defined 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 L_\infty-algebroids as that opposite to quasi-free differential graded algebras, identifying every L L_\infty-algebra with its dual Chevalley-Eilenberg algebra.

Specific examples

Opposite of SetSet and FinSetFinSet

The power set-functor

𝒫:Set opBool \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 opFinBoolAlg. FinSet^{op} \simeq FinBoolAlg \,.

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

References

For the definition in enriched category theory see page 12 of

  • Max Kelly, Basic concepts of enriched category theory (pdf)

Revised on June 25, 2013 18:23:35 by knuton? (146.50.72.20)