nLab copower

Idea

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

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

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

The dual notion is that of powering.

Definition

Definition

Let $V$ be a closed monoidal category serving as the cosmos for enrichment.

In a $V$-enriched category $\mathbf{C}$, the copower of an object $c \in \mathbf{C}$ by an object $v \in V$ is an object $v \cdot c \in \mathbf{C}$ with a ($V$-enriched-)natural isomorphism

$$\mathbf{C}\big( v \cdot c,\, c' \big) \;\;\cong\;\; \mathbf{V}\big( v ,\, \mathbf{C}(c,c') \big)$$

where

• $\mathbf{C}(-,-)$ denotes the $V$-valued hom-object of $C$,

• $\mathbf{V}(-,-)$ denotes correspondingly the internal hom of $V$ (i.e. the hom-object with respect to its canonical self-enrichment).

Remark

(terminology)
Copowers are frequently called tensors and a $V$-category having all copowers is called tensored, while the word “copower” is reserved for the case $V=Set$. 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.

Copowers as colimits

Conical colimits

For a Set-enriched category $C$, a copower $v \cdot c$ is the colimit of the constant functor $v \to C$ on $c$ (where $v$ is viewed as a discrete category).

Weighted colimits

A copower $v \cdot c$ is the colimit of the functor $1 \to C$ picking out $c$ weighted by the presheaf $1^{op} \to V$ picking out $v$.

Properties

• As described above, copowers are a kind of weighted colimit. Conversely, all weighted colimits can be constructed from copowers together with conical colimits (i.e., ordinary $Set$-based colimits with an enhanced $V$-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 $V$ be a Bénabou cosmos.

• In $V$ regarded as a $V$-enriched category over itself, the copower is just the given tensor product of $V$.

• If $B$ is $V$-copowered and $A$ is $V$-small, then the $V$-enriched functor category $B^A$ is $V$-copowered.

• If $C$ is $V$-copowered and $B \to C$ is a $V$-reflective full embedding, then $B$ is also copowered.

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

Example

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

$$\cdot \,\colon\, Set \times C \to C$$

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

$$S \cdot b \coloneqq \coprod_{s \in S} b \,.$$

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

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

Example

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

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

  for each $x\in X$ and each $a,b\in A$, where $()$ is the unit of $Free(X\times A)$. Here, for each $x\in X$ and each $a\in A$, we write $x\cdot a$ for the equivalence class of $(x,a)$.

• The product given by concatenation, i.e. by
  $$(x\cdot a)\cdot_{X\cdot A}(y\cdot b) = (x\cdot a)(y\cdot b),$$

  for each $x\cdot a,y\cdot b\in X\cdot A$;

• The unit given by
  $$1_{X\cdot A}=x\cdot 1_A,$$

  which is independent of $x$, as $x\cdot 1_A=()=y\cdot 1_A$ for all $x,y\in A$.

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

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

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

for each $x\in X$.

The copower $\cdot\colon Set\times Mon\to Mon$ also endows $Mon$ with skew monoidal structures $\triangleleft$ and $\triangleright$, given by

$$\begin{aligned} A\triangleleft B &= |B| \cdot A,\\ A\triangleright B &= |A| \cdot B \end{aligned}$$

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

Related concepts

• powered and copowered category

• (∞,1)-copower

• power

• bitensoring

References

Textbook accounts:

• Max Kelly, Section 3.7 of: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp. (ISBN:9780521287029);

  republished as:

  Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 (tac:tr10, pdf)

• Francis Borceux, Section 6.5 of: Handbook of Categorical Algebra Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994)

  (doi:10.1017/CBO9780511525865

• Emily Riehl, §3.7 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

A discussion on the advantages of the terminology “copower” over the older terminology “tensor”:

• Categories mailing list discussion