Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory kan lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology newpage noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeMar 21st 2012

    I have incorporated Jonas’ comment into the text at pretopos, changing the definition to “a category that is both exact and extensive”, as this is sufficient to imply that it is both regular and coherent.

    • CommentRowNumber2.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 14th 2017
    • (edited Jul 14th 2017)

    Just a terminological comment for the benefit of future readers:

    in the Elephant, (pretopos)==(coherent,effective,positive). In pretopos, (pretopos)==(exact,extensive)\Leftrightarrow(ditto, with “coherent” added).

    The discrepancy between “effective” and “exact” is terminological: “effective” is just Johnstone’s synonym for “exact” (cf. Elephant p.24)

    This leaves the question whether (coherent,exact,positive)\Leftrightarrow(coherent,exact,extensive); this is true for the maximally-strong, terminological reason that (positive)==definitionally==(extensive). (cf e.g. here for more)

    [ I shied away from making a footnote on this in pretopos, since to achieve a maximally-clear comment like “This is sometimes synonymously stated as “effective and positive” one has to know whether it is true in general that (effective,positive)\Rightarrow(coherent), as is true when “effective,positive” is replaced with “exact,extensive”; the latter I did not stop to to try to ascertain, in particular since “positive” seems to usually only be discussed for categories which are assumed to be coherent in the first place ]

    • CommentRowNumber3.
    • CommentAuthorkevin.watkins
    • CommentTimeMay 12th 2018
    When I view the pretopos page, in some places I see "∞-pretopos" where it ought to be "$\Pi$-pretopos" and "?W-pretopos" where it ought to be "$\Pi$-$W$-pretopos".

    I wonder if the history of the page in the database actually got corrupted? In particular, some related concepts links at the bottom are affected, and I don't remember them looking like that before.
    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2018

    It’s a bug we know about, see here. Hopefully Richard Williamson will be able to fix it.

  1. Thank you very much for reporting this, Kevin. I have now fixed ΠW\Pi W-pretopos and hopefully Π\Pi-pretopos (if there were any occurrences of ’\infty-pretopos’ which genuinely referred to \infty-categories, these will now wrongly be Π\Pi-pretoposes as well, but I could not find any such occurrences). Please report any other occurrences of question marks and seemingly erroneous \infty-symbols that you come across!

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 14th 2018

    Thanks for doing these fixes Richard!

    The term “\infty-pretopos” is also used in the literature (at least, in the Elephant) for what we on the nLab call an infinitary (1-)pretopos. Apparently we didn’t mention that anywhere though! I’ve now added a note to pretopos.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)