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
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.
If $0$ is an initial object in $\mathbf{C}$, then $0\downarrow\mathbf{C}$ is isomorphic to $\mathbf{C}$.
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”.
The category of commutative algebras over a field $F$ is the coslice under $F$ of the category CRing of commutative rings.
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:
and $\lim F$ is uniquely characterized by $\lim (p F)$.
Over a morphism $\gamma : d \to d'$ in $D$ the limiting cone over $p F$ (which exists by assumption) looks like
By the universal property of the limit this has a unique lift to a cone in the under category $t/C$ over $F$:
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 $Q \to \lim p F$
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 $\lim F$.
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 $A \to (U F)-Alg$ 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.)
under category
Discussion in the generality of $(\infty,1)$-categories:
Last revised on January 26, 2024 at 19:53:56. See the history of this page for a list of all contributions to it.