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-categories 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-type-theory cohomology combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry 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 homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories 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 subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeNov 6th 2013
   • PermaLink
   Author: Fosco
   Format: MarkdownItexI would like to add to the page [[category of elements]] the following characterization: The category of elements of a functor $F\colon C\to \Set$ is equivalent to the subcategory of $\Fun ([0]\star C, \Set )$ which coincide with $F$ when restricted to $C$ and send [0] to the terminal object in $\Set$. This seems quite natural to me and I wasn't able to disprove it. Can you provide me either a counterexample or a reference for this result I can add to the page?

   I would like to add to the page category of elements the following characterization:

   The category of elements of a functor $F\colon C\to \Set$ is equivalent to the subcategory of $\Fun ([0]\star C, \Set )$ which coincide with $F$ when restricted to $C$ and send [0] to the terminal object in $\Set$. This seems quite natural to me and I wasn’t able to disprove it.

   Can you provide me either a counterexample or a reference for this result I can add to the page?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeNov 6th 2013
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexNo, that can't be correct. Suppose $C$ is a discrete two-object category, for instance.

   No, that can’t be correct. Suppose $C$ is a discrete two-object category, for instance.

3. • CommentRowNumber3.
   • CommentAuthorFosco
   • CommentTimeNov 6th 2013
   • (edited Nov 6th 2013)
   • PermaLink
   Author: Fosco
   Format: MarkdownItexI am sorry, but where do you see the flaw in this example? The category of elements of $F\colon \{0,1\}\to \Set$ is the discrete category with objects the elements of $F(0)\sqcup F(1)$; the category of functors blah blah is the category of spans $F(0) \leftarrow * \to F(1)$. To be honest I argued in general, but in this and in other toy examples it seems to work. *Edit*: Now I see by myself; both categories are discrete, but the second has $F0\times F1$ objects, where the first has $F0\sqcup F1$. Ok, nevermind. Thank you for your time.

   I am sorry, but where do you see the flaw in this example? The category of elements of $F\colon \{0,1\}\to \Set$ is the discrete category with objects the elements of $F(0)\sqcup F(1)$; the category of functors blah blah is the category of spans $F(0) \leftarrow * \to F(1)$. To be honest I argued in general, but in this and in other toy examples it seems to work.

   Edit: Now I see by myself; both categories are discrete, but the second has $F0\times F1$ objects, where the first has $F0\sqcup F1$. Ok, nevermind. Thank you for your time.