nLab tensorial costrength




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

XF(Y)F(XY)X \otimes F(Y) \to F(X \otimes Y)


F(X)YF(XY)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(XY)XF(Y)F(X \otimes Y) \to X \otimes F(Y)


F(XY)F(X)YF(X \otimes Y) \to F(X) \otimes Y

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

When VV is symmetric, the braiding bb 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(YZ))(XTY)ZT(X\otimes (Y\otimes Z)) \to (X \otimes T Y) \otimes Z agree.


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


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 opV^{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 revV^{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.)


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:

Last revised on May 3, 2023 at 11:29:48. See the history of this page for a list of all contributions to it.