Limits, colimits, and comma categories

Let $D:J\to\mathcal C$ be a diagram with small indexing category $J$. A cone with vertex $T$ consists of maps $\lambda_j:T\to D(j)$ satisfying $D(a)\lambda_j=\lambda_k$ for every arrow $a:j\to k$. A limit is a terminal object in the category of such cones. Thus each competing cone has a unique map from its vertex into the limit vertex that recovers every leg. The direction of this mediating map is part of the definition.

In Set, a limit is the subset of the product of all the diagram's sets consisting of compatible tuples. The equations imposed by diagram arrows explain why a pullback is usually smaller than the unrestricted product. Smallness of the index allows these set constructions, and Set has all small limits. This is not a statement that every category has all limits, or that a proper-class-indexed product can automatically be treated as a set.

A cocone consists of maps $\iota_j:D(j)\to Q$ such that $\iota_kD(a)=\iota_j$. A colimit is initial among cocones: every other cocone receives one uniquely determined map from $Q$. In Set, begin with the tagged disjoint union of diagram elements and impose the equivalence relation generated by all arrow identifications. This quotient ensures that compatible functions on the original pieces descend to a well-defined function on the glued object.

A pushout illustrates the distinction between gluing and intersecting. It starts with a span whose two legs share a source, then identifies the two images of every source element. A pullback instead starts with arrows sharing a target and keeps pairs with equal images. The words are dual, but their set formulas are different. Unique mediating maps prove their universal properties; counting surviving elements is only a useful check of the chosen construction.

For functors $F:\mathcal A\to\mathcal C$ and $G:\mathcal B\to\mathcal C$, an object of $(F\downarrow G)$ is a triple $(a,b,\alpha)$ with $\alpha:F(a)\to G(b)$. A morphism to $(a',b',\alpha')$ is a pair $r:a\to a'$, $s:b\to b'$ satisfying $G(s)\alpha=\alpha'F(r)$. Composition is componentwise, and the equation remains true by functoriality. The downward-arrow notation must be read together with this explicit typing.

The slice category $\mathcal C\downarrow X$ has arrows into $X$ as objects and has the identity of $X$ as a terminal object. The coslice $X\downarrow\mathcal C$ reverses that arrangement and has the identity as initial. More generally, let $\Delta:\mathcal C\to\mathcal C^J$ be the constant-diagram functor and regard $D$ as a functor from the terminal category into $\mathcal C^J$. Then cones form $(\Delta\downarrow D)$ and cocones form $(D\downarrow\Delta)$. This recovers both universal-object directions systematically.