nLab copower

Contents

Context

Enriched category theory

Limits and colimits

Contents

Idea

In a closed monoidal category CC the tensor product aba \otimes b and internal hom [b,c][b,c] are related by the defining natural isomorphism

C(ab,c)C(a,[b,c]). C(a \otimes b, c) \simeq C(a, [b,c]) \,.

The notion of copowering generalizes this to the situation where a category CC does not act on itself by tensors, but where another category VV acts on CC.

The dual notion is that of powering.

Definition

Definition

Let VV be a closed monoidal category. In a VV-enriched category CC, the copower of an object xCx\in C by an object kVk\in V is an object kxCk\odot x \in C with a natural isomorphism

C(kx,y)V(k,C(x,y)) C(k\odot x, y) \cong V(k, C(x,y))

where C(,)C(-,-) is the VV-valued hom-functor of CC and V(,)V(-,-) is the internal hom of VV.

Remark

Copowers are frequently called tensors and a VV-category having all copowers is called tensored, while the word “copower” is reserved for the case V=SetV=Set. However, there seems to be no good reason for making this distinction. Moreover, the word “tensor” is fairly overused, and unfortunate since a tensor (= a copower) is a colimit, while a cotensor (= power) is a limit.

Properties

  • Copowers are a special sort of weighted colimit. Conversely, all weighted colimits can be constructed from copowers together with conical colimits? (i.e., ordinary SetSet-based colimits with an enhanced VV-universal property, although the latter is automatic if powers also exist), assuming these exist. The dual limit notion of a copower is a power.

Examples

Example

Let VV be a Bénabou cosmos.

(e.g.Kelly 1982, Sec. 3.7)

Example

(copowering of small categories over set)
Every locally small category CC with all coproducts is canonically copowered over Set: the copowering functor

:Set×CC \otimes \,\colon\, Set \times C \to C

sends (S,b)(S,b) to the coproduct of |S||S|-many copies of bCb \in C:

Sb sSb. S \otimes b \coloneqq \coprod_{s \in S} b \,.

The defining natural isomorphism in this situation is then just the fact that the hom functor sends colimits in its first argument to limits:

C( sSb,c) sSC(b,c)Set(S,C(b,c)). C(\coprod_{s \in S} b , c) \simeq \prod_{s \in S} C(b,c) \simeq Set(S, C(b,c)) \,.

Example

(copowering of the category of monoids)
A particularly illuminating instance of Example occurs when CC is the category of monoids (or that of groups). In this case, the copower XAX\otimes A of a monoid AA by a set XX is the free product of XX copies of AA, which can more concretely be described as a “one-sided version” of the tensor product of commutative monoids. Indeed, XAX\otimes A is the monoid consisting of

  • The set given by the quotient of the free (noncommutative) monoid Free(X×A)Free(X\times A) on X×AX\times A by the congruence relation \sim generated by the relations
    (x,a)(x,b) (x,ab), (x,1 A) () \begin{aligned} (x,a)(x,b) &\sim (x,a b),\\ (x,1_A) &\sim () \end{aligned}

    for each xXx\in X and each a,bAa,b\in A, where ()() is the unit of Free(X×A)Free(X\times A). Here, for each xXx\in X and each aAa\in A, we write xax\otimes a for the equivalence class of (x,a)(x,a).

  • The product given by concatenation, i.e. by
    (xa) XA(yb)=(xa)(yb), (x\otimes a)\cdot_{X\otimes A}(y\otimes b) = (x\otimes a)(y\otimes b),

    for each xa,ybXAx\otimes a,y\otimes b\in X\otimes A;

  • The unit given by
    1 XA=x1 A,1_{X\otimes A}=x\otimes 1_A,

    which is independent of xx, as x1 A=()=y1 Ax\otimes 1_A=()=y\otimes 1_A for all x,yAx,y\in A.

Explicitly, xXA\coprod_{x\in X}A is isomorphic to the above monoid via the isomorphism sending (x 1a 1)(x na n)(x_{1}\otimes a_{1})\cdots(x_{n}\otimes a_{n}) to the element of xXA\coprod_{x\in X}A given by the word a 1 (x 1)a n (x n)a^{(x_{1})}_{1}\cdots a^{(x_{n})}_{n}, where a i (x i)a^{(x_{i})}_{i} is the element a ia_{i} in the x ix_{i}-th copy of AA in the expression xXA\coprod_{x\in X}A.

The universal property of the copower XAX\otimes A states that a morphism of monoids from XAX\otimes A to a monoid BB is the same data as a “left-bilinear” map of sets f:A×XBf\colon A\times X\to B, satisfying

f(ab,x) =f(a,x)f(b,x), f(1 A,x) =1 B \begin{aligned} f(a b,x) &= f(a,x)f(b,x),\\ f(1_A,x) &= 1_B \end{aligned}

for each xXx\in X.

The copower :Set×MonMon\otimes\colon Set\times Mon\to Mon also endows MonMon with skew monoidal structures \triangleleft and \triangleright, given by

AB =|B|A, AB =|A|B \begin{aligned} A\triangleleft B &= |B|\odot A,\\ A\triangleright B &= |A|\odot B \end{aligned}

for each A,BObj(Mon)A,B\in Obj(Mon), where |A||A| and |B||B| are the underlying sets of AA and BB. While monoids in CMonCMon with respect to the tensor product of commutative monoids are semirings, monoids in MonMon with respect to \triangleleft and \triangleright recover left and right near-semirings.

References

Textbook accounts:

Last revised on October 26, 2021 at 04:52:13. See the history of this page for a list of all contributions to it.