Colimit ($\mathrm{colim}$)

A colimit is a universal cocone under a diagram: every other cocone factors through it uniquely.

The dual universal family is a coend $Q$ with maps $i_C:T(C,C)\to Q$ satisfying $i_{C}T(f,1_C)=i_{D}T(1_D,f)$ on $T(D,C)$. Every compatible family to another object factors uniquely through $Q$. For small $\mathcal{C}$ in a cocomplete category, it is the coequalizer ${\coprod }_{f:C\to D}T(D,C)\rightrightarrows {\coprod }_{C}T(C,C)\to Q$.

The colimit of two arrows $A\to B$ and $A\to C$. A map from the pushout is uniquely a pair of maps from $B$ and $C$ agreeing on $A$. In modules it is $(B\oplus C)/\{(f(a),-g(a)):a\in A\}$.

For a finite diagram in a category $D:\mathcal{J}\to \mathcal{C}$, set $X={\coprod }_{j}D(j)$ and $Y={\coprod }_{a:i\to j}D(i)$. The two maps $s,t:Y\rightrightarrows X$ on the $a$-summand are the injection of $D(i)$ and the injection of $D(j)$ after $D(a)$. Their coequalizer is the colimit of $D$. The pair $[s,1_X],[t,1_X]:Y⨿X\rightrightarrows X$ is a reflexive pair with the same coequalizer. Thus finite coproduct in a category constructions and coequalizers of reflexive pairs suffice, including the empty coproduct for the empty diagram. Finite products cannot replace coproducts here: the poset with elements $0,a,b,u_0,u_1,\dots$, order $0<a,b<u_{n+1}<u_n$, and incomparable $a,b$ has finite meets and top $u_0$, hence finite categorical products. Every reflexive parallel pair is an equal pair and has its identity as coequalizer, but $a,b$ have no least upper bound and therefore no coproduct.

A pushout of $B\leftarrow A\to C$ is a colimit with morphisms $B\to P$ and $C\to P$ making the square commute and universal among such commuting pairs. In the Category of sets, form the disjoint union of $B,C$ and identify the two images of each element of $A$.

For homomorphisms $i_j:H\to G_j$, the pushout of groups is $(G_1\ast G_2)/⟨⟨i_1(h)i_2(h{)}^{-1}:h\in H⟩⟩$. It has the universal property for homomorphisms out of $G_1,G_2$ agreeing on $H$. When the maps $i_j$ are injective, it is the amalgamated free product, with the factors and common subgroup embedded.

A category is cocomplete when it has every small colimit.

A coproduct is a colimit of a discrete diagram: it has injections from each summand and every family of maps out of the summands extends uniquely across it.

A countable coproduct is indexed by the natural numbers. In the Category of sets it is the disjoint union ${\coprod }_{n\in \mathbb{N}}{A}_n$.

The coequalizer of parallel arrows $f,g:A\rightrightarrows B$ is a universal arrow $q:B\to Q$ satisfying $qf=qg$. Every other arrow equalizing $f$ and $g$ factors uniquely through $q$.

For parallel arrows $f,g:B\rightrightarrows C$, a split coequalizer consists of $q:C\to Q$ and arrows $s:Q\to C$, $t:C\to B$ with $qf=qg$, $qs=1_Q$, $ft=1_C$, and $gt=sq$. These equations prove the coequalizer property: if $uf=ug$, then $u=uft=ugt=usq$, so $us$ is the unique factor through $q$. Every functor preserves this split coequalizer because it preserves these equations.

For $G:\mathcal{D}\to \mathcal{C}$, a parallel pair in $\mathcal{D}$ is G-split if its image admits a split coequalizer in $\mathcal{C}$. Reflection of such coequalizers means that an existing equalizing arrow in $\mathcal{D}$ is a coequalizer whenever its image is the corresponding split coequalizer. This asserts a property of existing arrows; it does not assert that a lift exists for every base quotient.

A multicolimit of a diagram is a family of cocones containing one initial object from every connected component of its cocone category.

Limits of shape $\mathcal{I}$ commute with colimits of shape $\mathcal{J}$ when, for every $D:\mathcal{I}\times \mathcal{J}\to \mathcal{C}$, the canonical comparisonis an isomorphism whenever both sides exist.

For small $J,K$ and a functor $D:J\times K\to \mathcal{C}$ into a complete category, the two iterated categorical limits are canonically isomorphic. They have the same projections to every $D(j,k)$ and the same universal property. The result follows also from preservation of categorical limits by the limit functor.

If groups $G,H$ have a common nontrivial quotient $K$, act on $K$ by left multiplication through $G$ and right inverse multiplication through $H$. The actions commute. There are no fixed points of a group action for $G$, but the $H$ orbit set is a singleton. The comparison $A^G/H\to (A/H{)}^G$ is therefore the map from the empty set to a singleton, which is not an isomorphism.

For commuting actions of finite groups $G,H$ with coprime orders, the canonical map $A^G/H\to (A/H{)}^G$ is a bijection. It is always injective. On any $G$-stable $H$-orbit, all $G$-orbits have the same size, dividing both $|G|$ and the $H$-orbit size, hence dividing $|H|$. Coprimality forces that size to be one, proving surjectivity. Thus categorical limits indexed by $G$ commute with colimits indexed by $H$ in the Category of sets.

A category is filtered when every finite diagram in it admits a cocone. Equivalently, it is nonempty, every pair of objects maps to a common object, and every pair of parallel arrows becomes equal after postcomposition.

A filtered colimit in a category is a colimit whose indexing category is a small filtered category. In the Category of sets, filtered colimits commute with finite limits in sets. A filtered colimit of modules is an example of this general categorical construction.

For a diagram of abelian groups indexed by a nonempty directed poset, take the set colimit as classes $[i,x]$, where $[i,x]=[j,y]$ when their images agree at some common later index. Define $[i,x]+[j,y]=[k,D_{ik}x+D_{jk}y]$ at any common upper bound $k$. Directedness and functoriality prove independence of choices. Negation is $-[i,x]=[i,-x]$ and zero is the class of any zero. This is the unique group structure making the canonical maps homomorphisms, and every compatible group cocone factors by a unique homomorphism. Hence the forgetful functor creates these colimits, even when transition maps are not injective.

A category is weakly filtered when every finite connected diagram in it admits a cocone. Equivalently, each of its connected components is a filtered category.

In the Category of sets, filtered colimits commute with finite limits. Elements in a finite limiting diagram involve only finitely many representatives and finitely many equalities, all of which can be realized at one common stage of a filtered diagram.

In a filtered colimit in a category of sets, any finite zigzag of generating identifications can be moved to one common stage. The filtered category conditions supply common objects and equalize the finitely many competing arrows. Consequently finitely many representatives and finitely many equalities can be realized simultaneously at one stage. This is the elementwise reason filtered colimits commute with finite limits in sets.

Let $A_n=\{m\in \mathbb{N}:m\ge n\}$ with the inverse-system inclusions $A_{n+1}\hookrightarrow A_n$, and let $B_n=C_n=1$. Each pushout of $1\leftarrow A_n\to 1$ is a singleton because $A_n$ is nonempty. Yet $\underleftarrow{\lim }{A}_n=\varnothing$, so the pushout after taking limits is $1{\bigsqcup }_{\varnothing }1$, a two-element set. Thus cofiltered limits do not in general preserve pushouts.

A small category $\mathcal{J}$ is sifted when it is connected and its diagonal functor $\mathcal{J}\to \mathcal{J}\times \mathcal{J}$ is a final functor. Equivalently, colimits of shape $\mathcal{J}$ in the Category of sets preserve finite products.

A local state classifier of $\mathcal{C}$ is a colimit of the inclusion of the wide subcategory containing all objects and only monomorphisms into $\mathcal{C}$.