nLab
sesquicategory

Idea

A sesquicategory is a (strict) 2-category in which the interchange law need not hold.

Definition

Sesquicategories can be straightforwardly defined just as for strict 2-categories except with the interchange law left out. (In order for this to make sense, one has to spell out the definition explicitly enough that the interchange law is a separate axiom.) This means that composition in a sesquicategory cannot be functor $\mathrm{hom}(y,z)\times \mathrm{hom}(x,y)\to \mathrm{hom}(x,z)$. So sesquicategories are more usually defined as categories enriched in Cat, where the monoidal structure for the enrichment is not the usual cartesian product but the tensor adjoint to the ‘unnatural’ hom, in which the hom-category $[C,D]$ has morphisms given by $C$-indexed families of arrows of $D$ without any naturality requirement.

Alternatively, a sesquicategory may be given as a category $C$ together with a functor $H:C^{\mathrm{op}}\times C\to \mathrm{Cat}$ whose composite $\mathrm{ob}\circ H:C^{\mathrm{op}}\times C\to \mathrm{Cat}\to \mathrm{Set}$ with the underlying-set functor is equal to the hom functor of $C$. Because of the equivalence $[C,\mathrm{Cat}(D)]\simeq \mathrm{Cat}[C,D]$ (for finitely complete $D$), this is the same as saying that a sesquicategory is given by a category $C$ together with an internal category $H$ in $[C^{\mathrm{op}}\times C,\mathrm{Set}]$ whose object $H_0$ of objects is the hom functor ${\mathrm{hom}}_C$ of $C$.

Remarks

A strict premonoidal category is the same as a sesquicategory with exactly one object.

A Gray-category does not have an underlying strict 2-category, but it does have an underlying strict sesquicategory. Thus, if one wants to define Gray-computads, it is natural to work with “sesqui-computads” as a partway stage; see for instance Surface diagrams

The name ‘sesquicategory’ literally means $1\frac{1}{2}$-category, although strictly speaking they are actually more general than $2$-categories (which are of course more general than $1$-categories). However, one can also view a $2$-category or sesquicategory as a $1$-category with extra structure or stuff (the $2$-cells and their composition), and in this way sesquicategories are partway between $1$-categories and $2$-categories, with only one axiom left out. (A strict 2-category can be considered directly as a 1-category with additional 2-cells added; for a weak 2-category one has instead to consider its “underlying” 1-category to be its homotopy category obtained by identifying isomorphic 1-morphisms.)

The paper Stell (1994) shows the relation with rewriting.

The paper Brown (2010) shows how a sesquicategory arises from a whiskered category.

References

• this discussion on the categories list.

• Stell, J., Modelling term rewriting by sesquicategories. Technical report, University of Keele (TR94-02).

• Ross Street, Categorical structures, p. 529–577 in Handbook of Algebra, vol. 1, edited by M. Hazewinkel, Elsevier 1996.

• Ronnie Brown, Possible connections between whiskered categories and groupoids, Leibniz algebras, automorphism structures and local-to-global questions. J. Homotopy Relat. Struct., 5(1) (2010) 305–318.