nLab regular semicategory

Idea

The notion of a regular semicategory generalizes the notion of a regular algebra from ring theory to (semi-)category theory. Just like regular algebras are (in general) non-unital algebras that nevertheless “behave” in many respects like unital ones, regular semicategories are semicategories on the verge of being categories.

Note that regular semicategories that are categories are not (necessarily) regular categories in the usual sense. In this case there is a clash of terminology between category theory and algebra.

Preliminaries

For convenience let us first recall a couple of concepts

Definition

Let $\mathcal{G}$, $\mathcal{H}$ be semicategories. A morphism $F:\mathcal{G}\to\mathcal{H}$ of semicategories assigns to all objects $X\in\mathcal{G}$ an object $F(X)\in\mathcal{H}$ and to every morphism $f:X\to Y$ in $\mathcal{G}$ a morphism $F(f):F(X)\to F(Y)$ in $\mathcal{H}$ such that $F(f\circ g)=F(f)\circ F(g)$.

A natural transformation $\alpha:\mathcal{G}\Rightarrow\mathcal{H}$ consists of a family $\{\alpha_X:F(X)\to G(X)\}_{X\in|\mathcal{G}|}$ in $\mathcal{H}$ indexed by the objects of $\mathcal{G}$ such that for all $f:X\to Y$ in $\mathcal{G}$ the following diagram commutes:

$$\array{ F(X)& \overset{F(f)}{\longrightarrow} & F(Y) \\ {}_{\alpha_X}\downarrow & & \downarrow _{\alpha_Y} \\ G(X)& \overset{G(f)}{\longrightarrow} & G(Y) }$$

Definition

A presheaf on a semicategory $\mathcal{G}$ is a morphism of semicategories $F:\mathcal{G}^{op}\to Set$. The category $Prsh(\mathcal{G})$ has objects presheaves on $\mathcal{G}$ and morphisms the natural transformations and is the called the category of presheaves of the semicategory $\mathcal{G}$.

$Prsh(\mathcal{G})$ is indeed a category! Denoting $\overline{\mathcal{G}}$ the category resulting from $\mathcal{G}$ by adding the missing identity morphisms, it is easy to check that $Prsh(\mathcal{G})\simeq PrSh(\overline{\mathcal{G}})$ and that the latter coincides with the usual presheaf category hence $Prsh(\mathcal{G})$ is even a Grothendieck topos.

Given $\mathcal{G}$ there is also a Yoneda morphism $Y_\mathcal{G}:\mathcal{G}\to Prsh(\mathcal{G})$ defined on objects as usual by $X\mapsto Hom_\mathcal{G}({}_-,X)$. Since semicategories embed into categories only iff they are categories themselves it follows that $Y_\mathcal{G}$ is fully-faithful iff $\mathcal{G}$ is a category!

Definition

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Example

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Properties

The most striking result is that although for a general regular semicategory $\mathcal{G}$ regular presheaves will not be Yoneda presheaves and vice versa nevertheless the subcategories $RegPsh(\mathcal{G})$ and $YonPsh(\mathcal{G})$ are identical in an identity-and-unity of opposites in the sense of Lawvere i.e. both are equivalent and occur in an essential localization of $Prsh(\mathcal{G})$.

Theorem

Let $\mathcal{G}$ be a regular semicategory. The functor $k:RegPsh(\mathcal{G})\to Prsh(\mathcal{G})$ defined on objects by $F\mapsto Nat(Y_\mathcal{G}({}_-),F)$ is fully-faithful and part of an adjoint string

$$i\dashv j\dashv k:RegPsh(\mathcal{G})\hookrightarrow Prsh(\mathcal{G})$$

with $k$ identifying the regular presheaves with the Yoneda presheaves and $i$ identifying them with the presheaves that are colimits of representables.

For the proof see Moens et al. (2002, p.179).

Related entries

• semicategory

• semipresheaf

• unity-and-identity of opposites

• Yoneda lemma

References

Regular semicategories were introduced in

• M.-A. Moens, U. Bernani-Canani, F. Borceux, On regular presheaves and regular semi-categories , Cah. Top. Géom. Diff. Cat. XLIII no.3 (2002) pp.163-190. (numdam)

Their quantaloid-enriched theory is studied in

• Isar Stubbe, Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories , Cah. Top. Géom. Diff. Cat. XLVI no.2 (2005) pp.99-121. (numdam)

For the origins in algebra of the concept see

• F. Grandjean, E. M. Vitale, Morita equivalence for regular algebras , Cah. Top. Géom. Diff. Cat. XXXIX (1998) pp.137-153. (numdam)