nLab tensorial costrength

Idea

A tensorial costrength (or simply costrength) for a functor $F \colon V \to V$ on a monoidal category $V$ is, in essence, the dual of a tensorial strength. Whereas a strength provides a coherent natural transformation of the form:

$$X \otimes F(Y) \to F(X \otimes Y)$$

or

$$F(X) \otimes Y \to F(X \otimes Y)$$

(a left-strength or right-strength respectively), a costrength provides a coherent natural transformation of the form:

$$F(X \otimes Y) \to X \otimes F(Y)$$

or

$$F(X \otimes Y) \to F(X) \otimes Y$$

(a left-costrength or right-costrength respectively).

When $V$ is symmetric, the braiding $b$ allows to obtain a left-costrength from a right-costrength, and in this setting, one usually drops “left-” or “right-” and simply talks about tensorial costrengths.

More generally, a costrength (on a monoidal category which is not necessarily symmetric) is a left-costrength and a coright-strength such that the two induced maps $T(X\otimes (Y\otimes Z)) \to (X \otimes T Y) \otimes Z$ agree.

Definition

For now, see tensorial strength, whose definition may be appropriately dualised.

Terminology

The terminology “costrength” was introduced by Blute, Cockett & Seely (1997) for the concept described here. This terminology is consistent with the usual convention in category theory of prefixing a term by “co-” when it is the same concept in the opposite category $V^{op}$.

Unfortunately, in the same year, Power & Robinson also used the terminology “costrength” to describe a right-strength. A right-strength is a left-strength with respect to the reversal of the tensor product, i.e. a left-strength in the reverse monoidal category $V^{rev}$. While both usages can be found in recent literature, we prefer the terminology that is consistent with prior categorical terminology.

(Note that the notion of “cotensorial strength” from Kock (1972) is different from both concepts, and involves a costrength-like transformation involving the internal hom of a monoidal category, rather than the tensor.)

Related concepts

• tensorial strength
• strong monad
• commutative monad

References

• R. F. Blute, J. R. B. Cockett, and R. A. G. Seely. Categories for computation in context and unified logic. Journal of Pure and Applied Algebra 116.1-3 (1997): 49-98.

The following paper introduces a conflicting definition of “costrength”, which we call a right-strength:

• John Power, and Edmund Robinson. Premonoidal categories and notions of computation. Mathematical structures in computer science 7.5 (1997): 453-468.

The notion of cotensorial strength is mentioned briefly in:

• Anders Kock, Strong functors and monoidal monads, Arch. Math 23 (1972) 113–120 [doi:10.1007/BF01304852, pdf]