nLab
under category

Coslice (under) categories

Coslice (under) categories

Definition

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 \big\downarrow C &\to& pt \\ \big\downarrow && \big\downarrow^{\mathrlap{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.

Examples

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

  • If 00 is an initial object in C\mathbf{C}, then 0C0\downarrow\mathbf{C} is isomorphic to C\mathbf{C}.

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

Properties

Limits and colimits

Proposition

If CC is a category with all limits, then a limit in any of its under categories t/Ct/C is computed as a limit in the underlying category CC.

In detail:

Let F:Dt/CF \colon D \to t/C be any functor.

Then, the limit over pFp \circ F in CC is the image under the evident projection p:t/CCp \colon t/C \to C of the limit over FF itself:

p(limF)lim(pF) p(\lim F) \simeq \lim (p F)

and limF\lim F is uniquely characterized by lim(pF)\lim (p F).

Proof

Over a morphism γ:dd\gamma : d \to d' in DD the limiting cone over pFp F (which exists by assumption) looks like

limpF pF(d) pF(γ) pF(d) \array{ && \lim p F \\ & \swarrow && \searrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

By the universal property of the limit this has a unique lift to a cone in the under category t/Ct/C over FF:

t limpF pF(d) pF(γ) pF(d) \array{ && t \\ & \swarrow &\downarrow & \searrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

It therefore remains to show that this is indeed a limiting cone over FF. Again, this is immediate from the universal property of the limit in CC. For let tQt \to Q be another cone over FF in t/Ct/C, then QQ is another cone over pFp F in CC and we get in CC a universal morphism QlimpFQ \to \lim p F

t Q limpF pF(d) pF(γ) pF(d) \array{ && t \\ & \swarrow & \downarrow & \searrow \\ && Q \\ \downarrow & \swarrow &\downarrow & \searrow & \downarrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

A glance at the diagram above shows that the composite tQlimpFt \to Q \to \lim p F constitutes a morphism of cones in CC into the limiting cone over pFp F. Hence it must equal our morphism tlimpFt \to \lim p F, by the universal property of limpF\lim p F, and hence the above diagram does commute as indicated.

This shows that the morphism QlimpFQ \to \lim p F which was the unique one giving a cone morphism on CC does lift to a cone morphism in t/Ct/C, which is then necessarily unique, too. This demonstrates the required universal property of tlimpFt \to \lim p F and thus identifies it with limF\lim F.

Remark

One often says “pp reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if U:ACU: A \to C is monadic (i.e., has a left adjoint FF such that the canonical comparison functor A(UF)AlgA \to (U F)-Alg is an equivalence), then UU both reflects and preserves limits. In the present case, the projection p:A=t/CCp: A = t/C \to C is monadic, is essentially the category of algebras for the monad T()=t+()T(-) = t + (-), at least if CC admits binary coproducts. (Added later: the proof is even simpler: if U:ACU: A \to C is the underlying functor for the category of algebras of an endofunctor on CC (as opposed to algebras of a monad), then UU reflects and preserves limits; then apply this to the endofunctor TT above.)

Last revised on January 20, 2021 at 03:35:28. See the history of this page for a list of all contributions to it.