Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck 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 homotopy homotopy-theory homotopy-type-theory index-theory infinity integration-theory internal-categories k-theory kan lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative 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 string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorJonasFrey
   • CommentTimeJan 21st 2018
   • PermaLink
   Author: JonasFrey
   Format: MarkdownItexI'm a bit unhappy about the fact that the definition of [creation of limits](https://ncatlab.org/nlab/show/created+limit) on the nlab is different from the definition of MacLane and on Wikipedia. I understand that they both use `strict' definitions and the nlab wants a non-strict one, but the nlab definition doesn't even generalize the standard one. For the benefit of people who might be confused about this in the future, let me pinpoint the difference: The nlab says: > Let $F:C\to D$ be a functor and $J:I\to C$ a diagram. We say that $F$ creates limits for $J$ if $J$ has a limit whenever the composite $F\circ J$ has a limit, and $F$ both preserves and reflects limits of $J$. ... whereas a non-strict version of the standard (MacLane/Wikipedia) definition would be > Let $F:C\to D$ be a functor and $J:I\to C$ a diagram. We say that $F$ creates limits for $J$ if $J$ has a limit whenever the composite $F\circ J$ has a limit, and **in this case** $F$ both preserves and reflects limits of $J$. Thus, the nlab definition forbids the existence of limits in $C$ if the limit in $D$ doesn't exist, whereas the standard definition doesn't say anything in this case. Right? I'm not necessarily asking to change this, but I think that at least the difference should be emphasized more - I have to admit that I only noticed the difference after looking at the page for the third time :-) Are there other sources that use the ``nlab version'' of the definition? (strict or non-strict) There's a [related thread here](https://nforum.ncatlab.org/discussion/7024/lifted-limit/) but I thought I start a new one with the correct title.

   I’m a bit unhappy about the fact that the definition of creation of limits on the nlab is different from the definition of MacLane and on Wikipedia. I understand that they both use ‘strict’ definitions and the nlab wants a non-strict one, but the nlab definition doesn’t even generalize the standard one. For the benefit of people who might be confused about this in the future, let me pinpoint the difference:

   The nlab says:

   Let $F:C\to D$ be a functor and $J:I\to C$ a diagram. We say that $F$ creates limits for $J$ if $J$ has a limit whenever the composite $F\circ J$ has a limit, and $F$ both preserves and reflects limits of $J$.

   … whereas a non-strict version of the standard (MacLane/Wikipedia) definition would be

   Let $F:C\to D$ be a functor and $J:I\to C$ a diagram. We say that $F$ creates limits for $J$ if $J$ has a limit whenever the composite $F\circ J$ has a limit, and in this case $F$ both preserves and reflects limits of $J$.

   Thus, the nlab definition forbids the existence of limits in $C$ if the limit in $D$ doesn’t exist, whereas the standard definition doesn’t say anything in this case. Right?

   I’m not necessarily asking to change this, but I think that at least the difference should be emphasized more - I have to admit that I only noticed the difference after looking at the page for the third time :-)

   Are there other sources that use the “nlab version” of the definition? (strict or non-strict)

   There’s a related thread here but I thought I start a new one with the correct title.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJan 22nd 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexDoes the difference really matter that much though? My experience has been that one only ever really talks about creation of limits that exist, which is why the nLab definition makes more sense to me. (In what sense can something that doesn't exist be "created"?)

   Does the difference really matter that much though? My experience has been that one only ever really talks about creation of limits that exist, which is why the nLab definition makes more sense to me. (In what sense can something that doesn’t exist be “created”?)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 22nd 2018
   • (edited Jan 22nd 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn any case it might be good to add a comment to the page that highlights the subtlety

   In any case it might be good to add a comment to the page that highlights the subtlety

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJan 22nd 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexMaybe what we should do is first give the definition *assuming*, as a precondition, that the limit exists in the base, and then remark that there are two ways one might choose to extend this to the case when the limit doesn't exist.

   Maybe what we should do is first give the definition assuming, as a precondition, that the limit exists in the base, and then remark that there are two ways one might choose to extend this to the case when the limit doesn’t exist.

5. • CommentRowNumber5.
   • CommentAuthorJonasFrey
   • CommentTimeJan 22nd 2018
   • PermaLink
   Author: JonasFrey
   Format: MarkdownItexI just made an edit emphasizing the non-equivalence more, and stating MacLane's definition explicitly instead of the slightly hazy ``sometimes the definition is given differently'' (before seeing Mike's last comment). Feel free to change or rollback.

   I just made an edit emphasizing the non-equivalence more, and stating MacLane’s definition explicitly instead of the slightly hazy “sometimes the definition is given differently” (before seeing Mike’s last comment). Feel free to change or rollback.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 4th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexNobody objected to #4, so I went ahead and tried it.

   Nobody objected to #4, so I went ahead and tried it.