nLab under category

Definition

Given a category $C$ and an object $c \in C$, the under category (also called coslice category) $c \downarrow C$ (alternative notations include $C^{c/}$ and $c/C$ and sometimes, confusingly, $c\backslash C$) is the category whose

• objects are morphisms in $C$ starting at $c$; $c \to d$

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

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

$$\array{ c \big\downarrow C &\to& pt \\ \big\downarrow && \big\downarrow^{\mathrlap{pt \mapsto c}} \\ [I,C] &\stackrel{d_0}{\to}& C }$$

in Cat, where

• $I$ is the interval category $\{0 \to 1\}$;

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

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

• $pt$, 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 $c$.

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

The dual notion is an over category.

Examples

Example

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

Example

(pointed objects)

If $C$ has a terminal object $\ast \,\in\, C$, then the coslice $C^{\ast/}$ is known as the category of pointed objects in $C$. For instance:

• the category of pointed sets, is the coslice of Set under the singleton set,

• the category of pointed topological spaces is the coslice of Top under the point space,

• the category of pointed simplicial sets is the coslice of sSet under the 0-simplex.

If $C$ is a monoidal category with tensor unit $I \,\in\, C$, then the coslice $I \downarrow C$ is also known as the category of pointed objects in a monoidal category. For instance:

• the category of pointed abelian groups is the coslice of Ab under the additive group of integers,

• the category of pointed modules is the coslice of Mod under the additive module of the ground ring

• the category of bi-pointed sets is the coslice of $Set_*$, the category of pointed sets, under the boolean domain.

• the category of pointed endofunctors is a coslice of and endo-functor category under the identity functor.

Generally, for any $c \in C$ one may think of the $c$-coslice category as the category of “$c$-pointed objects”.

Example

The category of commutative algebras over a field $F$ is the coslice under $F$ of the category CRing of commutative rings.

Properties

Limits and Colimits

Proposition

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

In detail:

Let $F \colon D \to t/C$ be any functor.

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

$$p(\lim F) \simeq \lim (p F)$$

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

Proof

Over a morphism $\gamma : d \to d'$ in $D$ the limiting cone over $p F$ (which exists by assumption) looks like

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

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

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

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

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

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

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

Remark

One often says “$p$ 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 $U: A \to C$ is monadic (i.e., has a left adjoint $F$ such that the canonical comparison functor $A \to (U F)\text{-Alg}$ is an equivalence), then $U$ both reflects and preserves limits. In the present case, the projection $p: A = t/C \to C$ is monadic, is essentially the category of algebras for the monad $T(-) = t + (-)$, at least if $C$ admits binary coproducts. (Added later: the proof is even simpler: if $U: A \to C$ is the underlying functor for the category of algebras of an endofunctor on $C$ (as opposed to algebras of a monad), then $U$ reflects and preserves limits; then apply this to the endofunctor $T$ above.)

The analogue is not true for general colimits in an under category, but it is true for all connected colimits:

Proposition

The forgetful functor $t/C \longrightarrow C$ creates all connected colimits that $C$ admits.

For this and the previous statement see also for instance Riehl 2017 Prop. 3.3.8

Related concepts

• over-category

  • slice category

  • under category

  • over topos

• over (∞,1)-category,

  • model structure on an over category

  • over-(∞,1)-topos

References

For more see most of the references at category theory.

Textbook accounts:

• Francis Borceux. Ex. 1.2.7 in: Handbook of Categorical Algebra, Vol 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) [doi:10.1017/CBO9780511525858]

• Emily Riehl, Ex. 1.1.3 in: Category Theory in Context, Dover Publications (2017) [ISBN:9780486809038, pdf, webpage]

Discussion in the generality of $(\infty,1)$-categories:

• Jacob Lurie, Rem. 1.2.5.9 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press (2009) [pup:8957, pdf]