Definitions \ref{defn:product}, \ref{defn:terminal}, \ref{defn:equalizer}, \ref{defn:pullback}, and \ref{defn:inverse} dualize to define:
- A coproduct $\coprod_{j \in J} A_j$ is the colimit of a diagram $(A_j){j \in J}$ indexed by a discrete category $J$. The legs of the colimit cone $\iota{j’} \colon A_{j’} \to \coprod_{j \in J} A_j$ are referred to as coproduct injections, though in pathological cases these maps might not be monomorphisms (see Exercise \ref{exc:injection-mono}).
- An initial object is the colimit of the empty diagram.
- A coequalizer is a colimit of a diagram indexed by the parallel pair category $\bullet\rightrightarrows \bullet$. The coequalizer of a parallel pair of maps $f,g \colon A \rightrightarrows B$ is the universal map $h \colon B \to C$ with the property that $hf = hg$. The colimit cone
\(\xymatrix{A \ar@<.5ex>[r]^f \ar@<-.5ex>[r]_g & B \ar@{->>}[r]^h & C}\) is called a coequalizer diagram. Diagrams of this shape are also called forks.
- A pushout is a colimit of a diagram indexed by the poset category $\bullet \leftarrow \bullet \to \bullet$. Dualizing the convention introduced in Definition \ref{defn:pullback}, the symbol ``$\ulcorner$’’ indicates that a commutative square
\(\xymatrix{ A \ar[r]^f \ar[d]_g \ar@{}[dr]\mid(.8){\ulcorner} & B \ar[d]^k \\ C \ar[r]_h & P}\) is a pushout, i.e., is a colimit diagram. The pushout is the universal commutative square under the maps $f$ and $g$.
- The colimit of a diagram indexed by the ordinal $\bbomega$ is called a sequential colimit or direct limit. The colimit of a diagram
\(\xymatrix{ F_0 \ar[r] & F_1 \ar[r] & F_2 \ar[r] & F_3 \ar[r] & \cdots}\) frequently denoted by $\varinjlim F_n$, defines a diagram of shape $\bbomega +1$. The term ``direct limit’’ is sometimes also used to refer to colimits of any shape.