nLab
opposite category

The idea

Given a category $C$, we obtain the ‘opposite category’ $C^{\mathrm{op}}$ by turning all the arrows around. This idea is important in understanding duality. For example, a comonoid in $C$ is the same as a monoid in $C^{\mathrm{op}}$. See also: opposite 2-category.

Definition

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

Definition in enriched category theory

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

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

and composition given by

$$C^{\mathrm{op}}(b,c)\otimes C^{\mathrm{op}}(a,b):=C(c,b)\otimes C(b,a)\stackrel{\sigma }{\to }C(b,a)\otimes C(c,b)\stackrel{{\circ }_C}{\to }{C}_{c,a}=:C^{\mathrm{op}}(a,c).$$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 $j_a:I\to C^{\mathrm{op}}(a,a)$ are those of $C$ under the identification $C^{\mathrm{op}}(a,a)=C(a,a)$.

Remarks

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 $L_{\mathrm{\infty }}$-algebroids as that opposite to quasi-free differential graded algebras, identifying every $L_{\mathrm{\infty }}$-algebra with its dual Chevalley-Eilenberg algebra.

References

for the definition in enriched category theory see page 12 of

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