What's a good motivating example for the concept of a slice category?

Solution 1:

If $P$ is a partial order and $a \in P$, then the interval $[a,\infty]$ is nothing else than the slice category $a / P$. If $a,b \in P$ with $a \leq b$, then $[a,b] = (a / P)/b$.

A coloring of a set $X$ by $r$ colors $c=\{c_1,\dotsc,c_r\}$ is just a map $X \to \{c_1,\dotsc,c_r\}$. The slice category $\mathsf{Set}/c$ is the category of $c$-colored sets, which plays a role in combinatorics.

In algebraic topology we often need base points; for example in order to make sense of the fundamental group. A pointed topological space is a pair $(X,x_0)$ consisting of a topological space $X$ and a point $x_0 \in X$. But such a point is the same as a morphism $\{*\} \to X$. Therefore, the category of pointed topological spaces is just the slice category $\{*\} / \mathsf{Top}$.

If $R$ is a commutative ring, then the category of commutative $R$-algebras is (isomorphic to) the slice category $R / \mathsf{CRing}$.

The coproduct of two commutative $R$-algebras $A,B$ is $A \otimes_R B$, the coproduct of two pointed spaces $X,Y$ is $X \vee_{*} Y$ (wedge sum). Both results can be seen as special cases of the general fact that coproducts in a slice category $X / C$ are just pushouts in $C$ (over $X$).

In general category theory, slice categories are used to switch between various notions of universal properties. This is useful for example in order to deduce Freyd's Adjoint Functor Theorem from Freyd's criterion for the existence of initial objects.

In general, if $X$ is an object of a category $C$, then often one refers to $C / X$ as the category of objects over $X$. The mental image is the following: $$\begin{array}{c} Y \\ \downarrow \\ X \end{array}$$ Grothendieck suggested to think of these objects as "generalized fibrations". If $C$ is a category of geometric objects (e.g. schemes, stacks, topological spaces, topological manifolds, smooth manifolds, orbifolds), then we often look at a full subcategory of $C / X$ with some geometric assumptions on $Y \to X$ (e.g. local isomorphism aka sheaf, covering, fibration, vector bundle, principal bundle) and try to understand $X$ via this subcategory of objects over $X$.

If $K$ is a field, then $\mathsf{Vect}_K ~ / ~ K$, after removing the trivial homomorphisms $0 : V \to K$, is equivalent to the category of affine spaces over $K$ ("vector spaces which have forgotten their origin"). This description makes it quite easy to find limits and colimits of affine spaces.

Solution 2:

If $X$ is a set, the slice category $\textsf{Set}/X$ can be thought of as the category of $X$-indexed collections of sets, where an object $f:Y\to X$ corresponds to the $X$-indexed collection of fibers $\{Y_x = f^{-1}\{x\}\mid x\in X\}$, and a morphism from $f:Y\to X$ to $g:Z \to X$ corresponds to a map of sets $Y_x\to Z_x$ for each $x\in X$.

This leads nicely into the observation that the fiber product $Y\times_X Z$ is just the product in the slice category, and in the category of sets it corresponds to taking the product on each fiber: $\{Y_x\times Z_x\mid x\in X\}$.

Moreover, this intuition about the meaning of the slice category is useful in other contexts, e.g. in algebraic geometry thinking of a map of schemes $X\to S$ as an algebraic family indexed by the base scheme $S$. Or in model theory (since you're a logician), thinking of a definable map of definable sets $X\to Y$ as a definable family of definable sets indexed by $Y$. Up to definable isomorphism, this can be arranged to be a projection map, so when we project the set $\phi(M)$ defined by $\phi(\bar{x},\bar{y})$ onto the $\bar{y}$ coordinates, we get the family $\{\phi(M,\bar{b})\mid \bar{b}\in \exists \bar{x}\,\phi(\bar{x},M)\}$.

Solution 3:

From the perspective of categorical logic, slice categories are also interesting.

First, recall that the main principle of categorical logic is that categorical structures arise from various forms of logical calculi by having propositions as objects and proofs or provability (depending on whether the calculus includes proofs or not) as morphisms (up to some equivalence).

Passing to the slice category over an object $P$ then means studying the logical calculus in the context of $P$.

Consider the simplest example of propositional, intuitionistic logic. There, having propositions $P$ up to logical equivalence as objects and provability of $P\to Q$ as the morphism set $\text{Hom}(P,Q)$ yields the concept of a Heyting algebra: a skeletal Cartesian closed category in which any morphism set is either empty or a singleton.

In such a Heyting algebra ${\mathcal H}$, passing to the slice category ${\mathcal H}/I$ over (the class of) a proposition $I\in {\mathcal H}$ means looking at propositions $P\in{\mathcal H}$ only for which $\text{Hom}_{\mathcal H}(P,I)\neq\emptyset$, i.e. those $P$ where $(P\Rightarrow I)=1$ in ${\mathcal H}$ ("$P$ implies $I$"). This is the same as the Heyting algebra attached to the modified logical calculus which has the same propositions $P,Q,...$, but in which a proof of "$P$ implies $Q$" is now a proof of "$P$ implies $Q$ under hypothesis of $I$" (or $P\wedge I\Rightarrow Q$ or $I\vdash P\Rightarrow Q$ in sequent style) in the old calculus, as apposed to just $P\Rightarrow Q$ or $\emptyset\vdash P\Rightarrow Q$ before.

In general, properties of the slice categories of a given category correspond to which kind of logic can be interpreted in the category.

For example, one might generalize the propositional intuitionistic logic from above, which can be interpreted in any Heyting algebra, to intuitionistic first order predicate logic: The latter can be seen as propositional intuitionistic logic fibered over the possible contexts of free variables, with introduction of unused variables and quantification allowing for passage between these contexts, and correspondingly it can be interepreted roughly in any category ${\mathcal C}$ in which for any "context" $X\in{\mathcal C}$ the subobjects of $X$ form a Heyting algebra, and in which pullbacks $\pi_X^{\ast}: {\mathcal C}/_{X}\to{\mathcal C}/_{X\times Y}$ (introducing unused, free variables) has a left adjoint $\exists_x$ and a right adjoint $\forall_x$. For example, one might take ${\mathcal C}=\text{Set}$, which gives the usual semantics of first order logic in set-models, but also ${\mathcal C}$ any elementary topos, which gives the Kripke-Joyal semantics.

More generally, in a category ${\mathcal C}$ with finite limits in which for any $f: X\to Y$ the pullback functor $f^{\ast}: {\mathcal C}/_{Y}\to {\mathcal C}/_{X}$ has both left and right adjoints $\exists_f,\forall_f: {\mathcal C}/_{X}\to{\mathcal C}/_{Y}$ (a locally Cartesian closed category), you may interpret intuitionistic higher order logic.