under category

Coslice (under) categories


Given a category CC and an object cCc \in C, the under category (also called coslice category) cCc \downarrow C (also written c/Cc/C and sometimes, confusingly, c\Cc\backslash C) is the category whose

  • objects are morphisms in CC starting at cc; cdc \to d

  • morphisms are commuting triangles c d 1 d 2. \array{ && c \\ & \swarrow && \searrow \\ d_1 &&\to && d_2 } \,.

The under category cCc\downarrow C is a kind of comma category; it is the strict pullback

cC pt ptc [I,C] d 0 C \array{ c\downarrow C &\to& pt \\ \downarrow && \;\;\downarrow^{pt \mapsto c} \\ [I,C] &\stackrel{d_0}{\to}& C }

in Cat, where

  • II is the interval category {01}\{0 \to 1\};

  • [I,C][I,C] is the internal hom in Cat, which here is the arrow category Arr(C)Arr(C);

  • the functor d 0d_0 is evaluation at the left end of the interval;

  • ptpt, the point, is the terminal category, the 0th oriental, the 0-globe;

  • the right vertical morphism maps the single object of the point to the object cc.

The left vertical morphism cCCc \downarrow C \to C is the forgetful morphism which forgets the tip of the triangles mentioned above.

The dual notion is an over category.


  • Set *Set_*, the category of pointed sets, is the undercategory ptSetpt\downarrow Set, where pt{}pt \simeq \{\bullet\} is the singleton set.

  • The category of commutative algebras over a field FF is the category FF \downarrowCRing of commutative rings under FF.

Revised on January 8, 2016 08:30:36 by Todd Trimble (