nLab lex total category

Idea

Lex-total categories are the good notion of topos. (Ross Street, 1981 p.201)

A lex total category is a category whose Yoneda embedding is a localization thereby generalizing Giraud's characterization of Grothendieck toposes as localizations of presheaf toposes hence they can be viewed as a candidate for a notion of “topos”.

Definition

Recall that a locally small category $\mathcal{C}$ is called total if the Yoneda embedding $Y:\mathcal{C}\hookrightarrow Set^{\mathcal{C}^{op}}$ has a left adjoint $L$.

Definition

A locally small category $\mathcal{C}$ is called lex total if the Yoneda embedding $Y:\mathcal{C}\hookrightarrow Set^{\mathcal{C}^{op}}$ has a left exact (=finite limit preserving) left adjoint $L$.

Examples

In particular, totally distributive categories where $L$ has a further left adjoint $W$ are lex total.

Below we will see that all Grothendieck toposes are lex total.

Properties

Theorem

A lex total category $\mathcal{E}$ is a Grothendieck topos iff $\mathcal{E}$ has a small set of generators.

The result was announced by Walters on the Isle of Thorns in 1976, a proof can be found in Street (1981, p.206). More detailed information on this characterization in particular concerning the size issues involved and the algebraic perspective it avails can be found at Grothendieck topos or the blog posts by Bob Walters.

Related entries

• total category

• totally distributive category

• infinitary pretopos

Link

• Bob Walters, Lex total categories and Grothendieck toposes I-IV , series of blog posts 2014. (link)

References

• Rory Lucyshyn-Wright, Totally distributive toposes , arXiv:1108.4032 (2011). (abstract)

• Ross Street, The family approach to total cocompleteness and toposes , Trans. A. M. S. 284 (1978) pp.355-369.

• R. Street, Notions of topos , Bull. Austral. Math. Soc. 23 no.2 (1981) pp.199-208.

• R. Street, R. Walters, Yoneda structures on 2-Categories , JA 50 no.2 (1978) pp.350-379.

• R. J. Wood, Some remarks on total categories , JA 75 2 (1982) 538-545 [doi:10.1016/0021-8693(82)90055-2]