regular semicategory

- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- 2-category, (2,1)-category
- 1-category
- 0-category
- (?1)-category?
- (?2)-category?

- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

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 aclash of terminologybetween category theory and algebra.

For convenience let us first recall a couple of concepts

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:

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

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!

…

…

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 $Prsh(\mathcal{G})$.

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).

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)

Last revised on May 30, 2018 at 05:04:53. See the history of this page for a list of all contributions to it.