# [ct/1] Natural Transforms, Products, Free Categories

Created: Wed, 26 Sep 2018Last updated: Fri, 22 Feb 2019

For the product set \(X \times Y\), given \(W, f, g\), and projections \(p : X \times Y \mapsto X\), \(q : X \times Y \mapsto Y\), the following diagram commutes:

\[ \begin{xy} \xymatrix{ & W \ar@{.>}[d]|{\exists! h} \ar[dl]_f \ar[dr]^g & \\ X & X \times Y \ar[l]_p \ar[r]^q & Y } \end{xy} \]

The unique function \(h\) is given by \(hw = \langle fw, gw \rangle\) for every \(w \in W\).

The correspondence \(h \mapsto \langle ph, qh \rangle = \langle f, g \rangle\) is given by the bijection: \[\text{hom}(W, X \times Y) \cong \text{hom}(\langle W, W \rangle, \langle X, Y \rangle)\]

We write \(\text{hom}(\langle U, V \rangle, \langle X, Y \rangle)\) for the pair of arrows \(U \rightarrow X\) and \(V \rightarrow Y\). The construction involves two additional constructions: \(\Delta W = \langle W, W \rangle\) and \(\langle X, Y \rangle \rightarrow X \times Y\). \(\Delta\) is said to be `left adjoint` to the product \(X \times Y\).

The one-point object \(1 = \{0\}\) serves as identity for the "cartesian product" operation: \[1 \times X \xrightarrow{\lambda} X \xleftarrow[]{\rho} X \times 1\]

given by \(\lambda : \langle 0, x \rangle \rightarrow x\), \(\rho : \langle x, 0 \rangle \rightarrow x\).

## Monoid and group

A monoid is a set \(M\) equipped with \(\mu : M \times M \rightarrow M\) and \(\eta : 1 \rightarrow M\) such that the following diagrams commute:

\[ \begin{xy} \xymatrix{ M \times M \times M \ar[r]^{1 \times \mu}\ar[d]^{\mu \times 1} & M \times M \ar[d]^{\mu} \\ M \times M \ar[r]^{\mu} & M } \end{xy} \]

\[ \begin{xy} \xymatrix{ 1 \times M \ar[r]^{\eta \times 1}\ar[d]^\lambda & M \times M \ar[d]^\mu & M \times 1 \ar[l]_{1 \times \eta}\ar[d]^\rho \\ M & M & M } \end{xy} \]

This means that \(\mu \circ (\eta \times 1) = \lambda\) and \(\mu \circ (1 \times \eta) = \rho\).

Writing it out in terms of the elements, we get the familiar associativity axioms:

\[ \begin{xy} \xymatrix{ \langle x, y, z \rangle \ar@{|->}[r]\ar@{|->}[d] & \langle xy, z \rangle \ar@{|->}[d] \\ \langle x, yz \rangle \ar@{|->}[r] & x(yz) = xy(z) } \end{xy} \]

and the identity axioms:

\[ \begin{xy} \xymatrix{ \langle 0, x \rangle \ar@{|->}[r]\ar@{|->}[d] & \langle u, x \rangle \ar@{|->}[d] \\ x \ar@{}[r]|{=} & ux } \end{xy} \begin{xy} \xymatrix{ \langle x, u \rangle \ar@{|->}[d] & \langle x, 0 \rangle \ar@{|->}[l]\ar@{|->}[d] \\ xu \ar@{}[r]|{=} & x } \end{xy} \]

Then a category can be defined as a monoid with "product over \(O\)": \[A \times_0 A = \{\langle f, g \rangle \mid f, g \in A; \text{dom } f = \text{cod } g\}\]

We describe a group as a monoid with the additional operation \(\eta : M \rightarrow M\) corresponding to \(x \rightarrow x^{-1}\):

\[ \begin{xy} \xymatrix{ M \ar[d]\ar[r]^\delta & M \times M \ar[r]^{1 \times \eta} & M \times M \ar[d]^\lambda \\ 1 \ar[rr] & & M } \end{xy} \begin{xy} \xymatrix{ x \ar@{|->}[d]\ar@{|->}[r] & \langle x, x \rangle \ar@{|->}[r] & \langle x, x^{-1} \rangle \ar@{|->}[d] \\ 0 \ar@{|->}[rr] & & xx^{-1} = u } \end{xy} \]

Equivalently, a group is a category with one object, and in which every arrow has a two-sided inverse under composition.

\(\Delta\) is a category with objects all finite ordinals, and arrows all order-preserving functions. From algebraic topology, it is called the `simplicial category`.

## Functor

A functor is a morphism of categories. Let \(T : B \rightarrow C\) be a functor of two categories. Then, the object function assigns to each object \(b \in B\), an object \(Tb \in C\); the arrow function assigns each arrow \(b \rightarrow b' \in B\) to the arrow \(Tb \rightarrow Tb' \in C\), in such a way that: \[T_c(1) = 1_{Tc} \quad T(f \circ g) = Tf \circ Tg\]

the latter whenver \(f \circ g\) is defined in \(B\).

They arise naturally in algebraic topology, where we have a topological space \(X\) lifted to a [homology group](/at/1#homology) \(H_n(X)\); topological spaces map to abelian groups as \(A : \textbf{Top} \rightarrow \textbf{Ab}\). In algebra, a forgetful functor would erase the group structure yielding \(B : \textbf{Grp} \rightarrow \textbf{Set}\).

A functor is full when it is surjective, faithful when it is injective, fully faithful when it is bijective, in set-theoretic terms.

Functors also satisfy the law of associativity of compositions, just like the category does.

## Subcategory

A subcategory \(S\) of category \(C\) includes some of the arrows in \(C\), and:

- For each arrow \(f\), the objects \(\text{dom } f, \text{codom } f\)
- For each object \(s\), the identity arrow
- For each pair of arrows \(s \rightarrow s' \rightarrow s''\), the composition of arrows

## Natural transformation

A natural transformation is a morphism of functors. Given two functors \(S, T : B \rightarrow C\), let \(\tau : S \overset{\bullet}{\rightarrow} T\) be an NT. Then, every arrow \(f : b \rightarrow b' \in S\) is mapped to \(\tau_b = \tau b : Sb \rightarrow Sb' \in C\), so the following diagram commutes:

\[ \begin{xy} \xymatrix{ Sb \ar[r]^{\tau_b}\ar[d]^{Sb} & Tb \ar[d]^{Tf} \\ Sb' \ar[r]^{\tau_{b'}} & Tb' } \end{xy} \]

If we think of \(S\) as giving a picture in \(B\) of all objects and arrows in \(C\), we get the following commutative diagrams:

\[ \begin{xy} \xymatrix{ a \ar[dd]^h\ar[dr]^f & \\ & b \ar[dl]^g \\ c & } \end{xy} \begin{xy} \xymatrix{ Sa \ar[rr]^{\tau_a}\ar[dr]^{Sf}\ar[dd]^{Sh} & & Ta \ar[dr]^{Tf}\ar[dd]|{\hole} & \\ & Sb \ar[rr]^<<<<<{\tau_b}\ar[dl]^{Sg} & & Tb \ar[dl]^{Tg} \\ Sc \ar[rr]^{\tau_c} & & Tc & } \end{xy} \]

\(\tau_a, \tau_b, \tau_c\) are called the `components of the NT`.

The [determinant](/la#determinant) is an NT \(\tau : \textbf{CRng} \overset{\bullet}{\rightarrow} \textbf{Grp}\). Let \(\text{det}_K M\) be the determinant of matrix \(M\) with entries in commutative ring \(K\). \(det_K\) is a morphism \(\text{GL}_n K \overset{\bullet}{\rightarrow} K^*\) of groups.

\[ \begin{xy} \xymatrix{ \text{GL}_n K \ar[r]^{\text{det}_K}\ar[d]^{\text{GL}_n f} & K^* \ar[d]^{f^*} \\ \text{GL}_n K' \ar[r]^{\text{det}_{K'}} & K'^{*} } \end{xy} \]

## Monics, epis, and zeros

Arrow \(e : a \rightarrow b \in C\) is invertible if there exists \(e' : b \rightarrow a \in C\) with \(e'e = 1_a, ee' = 1_b\). If \(e'\) is unique, \(e' = e^{-1}\), and the isomorphism \(a \cong b\) holds.

For arrow \(m : a \rightarrow b\), \(m \circ a = m \circ b \implies a = b\) is the left cancelation property, and \(m\) is termed `monic`. \(a \circ m = b \circ m \implies a = b\) is the right cancelation property, and \(m\) is termed `epi`. In \(\mathbf{Set}\), monic arrows are injections and epi arrows are surjections.

For arrow \(h : a \rightarrow b\), arrow \(r : b \rightarrow a\) is right inverse if \(hr = 1_b\); \(r\) is termed as the `section` of \(h\). Similarly, for \(l : b \rightarrow a\), \(lh = 1_a\), \(l\) is termed as the `retraction` of \(h\).

If, for each object \(a \in C\), there exists exactly one arrow \(a \rightarrow t\), \(t\) is termed as the `terminal object`. On the other hand, for each object \(a \in C\), there exists exactly one arrow \(i \rightarrow a\), \(i\) is termed as the `initial object`. In \(\mathbf{Set}\), the empty set is initial, and any one-point set is terminal. An object which is both initial and final is called the `null object` \(z \in C\); then there exists a unique arrow \(a \rightarrow z \rightarrow b\), for any two objects \(a, b \in C\) called the `zero arrow`.

A groupoid is a category in which all morphisms are invertible. An example would be the fundamental groupoid \(\pi(X)\), from algebraic topology. A groupoid is said to be connected if there is an arrow between any two objects in the groupoid.

## Examples

- \(\textbf{Rng}\) is the category of all small rings
- \(\textbf{Top}\) is the category of topological spaces, where objects are the set of small topological spaces, and morphisms are continuous maps between spaces
- \(\textbf{Toph}\) is the category of topological spaces, where objects are the set of small topological spaces, and morphisms are homotopies
- \(\textbf{Set}_*\) is the small category of pointed sets
- \(\textbf{Top}_*, \textbf{Toph}_*\) are the categories of small pointed topological spaces; in the latter, morphisms are homotopies with a fixed basepoint \(*\)
- \(\mathbf{R-Mod}\) is the category of small left R-modules
- \(\mathbf{Vec}_F\) is the category of vector spaces
- \(\mathbf{Rel}\) is the category of small sets, where morphisms are binary relations
- A concrete category is the pair \(\langle C, U \rangle\), where \(C\) is a category, and \(U\) is a faithful functor \(U : C \rightarrow \textbf{Set}\)

## Hom-Sets

\(\text{hom}(a, b)\) is used to denote the set of all morphisms from \(a\) to \(b\). A small category can be defined in terms of hom-sets, with the following data:

- Objects \(a, b, c, \ldots\)
- For every pair of objects \(\langle a, b \rangle\), a function associating \(\text{hom}(a, b)\) to the pair
- For every triple \(\langle a, b, c \rangle\), the composition axiom:

\[ \begin{xy} \xymatrix{ \text{hom}(c, d) \times \text{hom}(b, c) \times \text{hom}(a, b) \ar[r]\ar[d] & \text{hom}(b, d) \times \text{hom}(a, b) \ar[d] \\ \text{hom}(c, d) \times \text{hom}(a, c) \ar[r] & \text{hom}(a, d) } \end{xy} \]

- For every object \(b\), \(1_b \in \text{hom}(b, b)\); the identity axiom

An Ab-category or preaddive category is a category in which hom-sets are additive abelian groups, and composition is bilinear: given \(f, f' : a \rightarrow b\), \(g, g' : b \rightarrow c\), \[(f + f') \circ (g + g') = f \circ g + f \circ g' + f' \circ g + f' \circ g'\]

\(\textbf{Ab}, \textbf{R-Mod}, \textbf{Mod-R}\) are all Ab-categories. \(\textbf{Ab-cat}\) is used the denote the category of all small Ab-categories.

## Duality

We define \(C^{op}\) to be the opposite category, with the same objects as \(C\), and directions of arrows reversed. For each \(f : a \rightarrow b \in C\), is an \(f^{op} : b \rightarrow a \in C^{op}\). Also \(f^{op} g^{op} = (gf)^{op}\). The assignments \(A \mapsto A^{op}\) and \(F \mapsto F^{op}\) define a covariant functor \(\textbf{Cat} \rightarrow \textbf{Cat}\).

The opposite functor \(F^{op} : B^{op} \rightarrow A^{op}\) is defined in the usual way. A functor \(S : A^{op} \rightarrow B\) can be defined in terms of \(A\) as \(\bar{S}\), a contravariant functor. Then, \(\bar{S}(m \circ n) = \bar{S}(n) \circ \bar{S}(m)\).

Hom-sets provide an example of covariant and contravariant functors. For each object \(a \in C\), the object function of the covariant hom-functor sends: \[C(a, -) = \text{hom}(a, -) : C \rightarrow \mathbf{Set}\]

and the arrow function sends each arrow \(k : b \rightarrow b'\) to: \[\text{hom}(a, k) : \text{hom}(a, b) \rightarrow \text{hom}(a, b')\]

For each object \(b \in C\), the object function of the contravariant hom-functor sends: \[C(-, b) = \text{hom}(-, b) : C^{op} \rightarrow \mathbf{Set}\]

and the arrow function sends each arrow \(g : a \rightarrow a'\) to: \[\text{hom}(g, b) : \text{hom}(a', b) \rightarrow \text{hom}(a, b)\]

To simplify the noation, introduce \(k_*\) for "composition with k on the left" or "composition induced by k", and \(g^*\) for "composition with g" on the right: \[k_* f = k \circ f \quad g^* f = f \circ g\]

For \(k : a \rightarrow a'\) and \(g : b \rightarrow b'\), we have:

\[ \begin{xy} \xymatrix{ \text{hom}(a', b) \ar[r]^{k^*}\ar[d]^{g_*} & \text{hom}(a, b) \ar[d]^{g_*} \\ \text{hom}(a', b') \ar[r]^{k^*} & \text{hom}(a, b') } \end{xy} \]

## Product structure

\(P_1\) along with \(X\) and \(Y\) defines a product. There exists a unique morphism between \(P_1\) and \(P_2\) preserving the product structure.

\[ \begin{xy} \xymatrix{ P_2 \ar@/_/[ddr]_N \ar@{.>}[dr]|{\exists!} \ar@/^/[drr]^M \\ & P_1 \ar[d]^Q \ar[r]_P & X \\ & Y & } \end{xy} \]

\(P, Q\) are functors called `projections` of \(P_1\) onto \(X\) and \(Y\). Elements of \(P_1\) are pairs \(\langle x, y \rangle\).

Let \(U : X \rightarrow Y, V : X' \rightarrow Y'\) be two functors; then \(U \times V : X \times Y \rightarrow X' \times Y'\) is the product of two functors, and we yield the following commutative diagram:

\[ \begin{xy} \xymatrix{ X \ar[d]^{U} & X \times Y \ar[l]_P\ar[r]^Q\ar@{.>}[d]|{U \times V} & Y \ar[d]^{V} \\ X' & X' \times Y' \ar[l]_{P'}\ar[r]^{Q'} & Y' } \end{xy} \]

Moreover, when \(U \circ U'\) and \(V \circ V'\) are defined, we have \((U \times U') \circ (V \times V') = UU' \times VV'\), making \(\times\) into a functor: \[\times : \textbf{Cat} \times \textbf{Cat} \rightarrow \textbf{Cat}\]

Functor \(S : B \times C \rightarrow D\) is defined to be a bifunctor or a functor in two variable objects. Hom-sets may be described as bifunctors \(\text{hom} : C^{op} \times C \rightarrow \textbf{Set}\).

Consider NT \(\alpha\) between bifunctors \(S, S' : B \times C \rightarrow D\): \(\alpha(b, c) : S(b, c) \rightarrow S'(b, c)\) is called `natural` in \(b\) if: \[\alpha(-, c) : S(-, c) \overset{\bullet}{\rightarrow} S'(-, c)\]

Consider the product category \(C \times \textbf{2}\), where \(\textbf{2}\) is the category of the two-point set \(\{0, 1\}\):

\[ \begin{xy} \xymatrix{ & & \bullet \ar[dr] & \\ C \times 1 & \bullet \ar[rr]\ar[ur] & & \bullet \\ & & \bullet \ar[dr]\ar@{.>}[uu] & \\ C \times 0 & \bullet \ar[rr]\ar[ur]\ar[uu] & & \bullet \ar[uu] } \end{xy} \]

Here \(T_0, T_1 : C \rightarrow C \times \mathbf{2}\) are functors from "bottom" to "top". \(T_0 f = \langle f, 0 \rangle\), \(T_1 f = \langle f, 1 \rangle\). If \(\uparrow : 0 \rightarrow 1\) denotes the unique non-identity arrow in \(\mathbf{2}\), then \(T_0, T_1\) are defined by: \[\mu : T_0 \overset{\bullet}{\rightarrow} T_1 \quad \mu c = \langle c, \uparrow \rangle\]

for any object \(c\). It maps "bottom" to "top", and is natural. We call \(\mu\) the universal natural transform from \(C\).

## Functor category

Consider functors \(R, S, T : C \rightarrow B\) and NTs \(\sigma : R \overset{\bullet}{\rightarrow} S, \tau : S \overset{\bullet}{\rightarrow} T\). Then, \((\tau \bullet \sigma) c = \tau c \circ \sigma c\), and \(\tau \bullet \sigma : R \overset{\bullet}{\rightarrow} T\), and we get the following commutative diagram:

\[ \begin{xy} \xymatrix{ Rc \ar[r]^{Rf}\ar[d]^{\sigma_c}\ar@/_2pc/[dd]_{(\tau \bullet \sigma) c} & Rc' \ar[d]^{\sigma_{c'}}\ar@/^2pc/[dd]^{(\tau \bullet \sigma) c'} \\ Sc \ar[r]^{Sf}\ar[d]^{\tau_c} & Sc' \ar[d]^{\tau_{c'}} \\ Tc \ar[r]^{Tf} & Tc' } \end{xy} \]

The category of functors may be written as \(B^C = \text{Funct}(C, B)\), with objects the functors \(T : C \rightarrow B\), and morphisms the NTs between two such functors. The hom-set for this category is: \[\text{Nat}(S, T) = B^C(S, T) = \{\tau \mid \tau : S \overset{\bullet}{\rightarrow} T\}\]

If \(B = \{0, 1\}\), then \(B^1\) is isomorphic to \(B\), and \(B^2\) is the category whose objects are arrows \(f : a \rightarrow b\) of \(B\), and whose arrows \(f \rightarrow f'\) are those that, along with pair \(\langle h, k \rangle\), make the following diagram commute:

\[ \begin{xy} \xymatrix{ a \ar[r]^h\ar[d]^f & a' \ar[d]^{f'} \\ b \ar[r]^k & b' } \end{xy} \]

## Category of all categories

Natural transforms satisfy the interchange law with respect to horizontal and vertical products:

\[ \begin{xy} \xymatrix{ A \ar@<-2ex>[r] \ar[r]_{\downarrow \tau} \ar@<2ex>[r]_{\downarrow \sigma} & B \ar@<-2ex>[r]\ar[r]_{\downarrow \tau'} \ar@<2ex>[r]_{\downarrow \sigma'} & C } \end{xy} \]

\[(\tau' \bullet \sigma') \circ (\tau \bullet \sigma) = (\tau' \circ \tau) \bullet (\sigma' \circ \sigma)\]

## Comma category

If \(b\) is an object of category \(C\), \((b \downarrow C)\) is a category in which objects are objects of \(\langle f, c \rangle\), where \(f : b \rightarrow c\), and arrows \(h : \langle f, c \rangle \rightarrow \langle f', c' \rangle\), such that \(h \circ f = f'\):

\[ \begin{xy} \xymatrix{ & b \ar[ld]_f \ar[rd]^{f'} & \\ c \ar[rr]^h & & c' } \end{xy} \]

Example: If \(\mathbf{*}\) denotes a one-point set, \((\mathbf{*} \downarrow \mathbf{Set})\) denotes the category of pointed sets.

If \(b\) is an object of category \(C\), and \(S : D \rightarrow C\) a functor, the category \((b \downarrow S)\) of objects under \(b\) has as objects pairs \(\langle f, d \rangle\), where \(d \in \text{Obj } D\) and \(f : b \rightarrow Sd\), and arrows \(h : \langle f, d \rangle \rightarrow \langle f', d' \rangle\) all those arrows \(h : f \rightarrow f'\) such that \(Sh \circ f = f'\):

\[ \begin{xy} \xymatrix{ & b \ar[ld]_f \ar[rd]^{f'} & \\ Sd \ar[rr]^{Sh} & & Sd' } \end{xy} \]

Consider \(C \xleftarrow{T} A \xrightarrow{S} B\) where \(S\) and \(T\) are NTs. Then the category \((T \downarrow S)\), also written as \((T, S)\) is called the comma category. It has as objects triples \(\langle c, b, f \rangle\), and as arrows \(\langle c, b, f \rangle \rightarrow \langle c', b', f' \rangle\) all pairs \(\langle k, h \rangle\) of arrows \(k : c \rightarrow c'\), \(h : b \rightarrow b'\) such that \(f' \circ Tk = Sh \circ f\):

\[ \begin{xy} \xymatrix{ Tc \ar[r]^{Tk}\ar[d]_f & Tc' \ar[d]^{f'} \\ Sb \ar[r]^{Sh} & Sb' } \end{xy} \]

We get the following diagram when we consider projections of the comma category:

\[ \begin{xy} \xymatrix{ & & \langle c, b, f : Tc \rightarrow Sb \rangle \ar@{|->}[d]\ar@{|->}[lld]\ar@{|->}[rrd] & & \\ b \ar@{|->}[r] & Sb & (f : Tc \rightarrow Sb) \ar@{|->}[l]\ar@{|->}[r] & Tc & c \ar@{|->}[l] } \end{xy} \]

## Graphs and free categories

A graph may be pictured, like a category, with objects as vertices and arrows as edges, except that there are no identity or composite arrows; for this reason, it may be called a `precategory`.

A directed graph is a set of objects \(O\) and a set \(A\) of arrows \(f\) such that there exists a pair of arrows:

\[ \begin{xy} \xymatrix{ A \ar@<-1ex>[r]_{\partial_1} \ar@<1ex>[r]^{\partial_0} & O & \partial_0 f = \text{dom } f & \partial_1 f = \text{codom } f } \end{xy} \]

A morphism between graphs \(D : G \rightarrow G'\) is a pair of functions \(D_O : O \rightarrow O'\) and \(D_A : A \rightarrow A'\) such that: \[D_O \partial_0 f = \partial_0 D_A f \quad D_O \partial_1 f = \partial_1 D_A f\]

A forgetful functor \(U : \textbf{Cat} \rightarrow \textbf{Grph}\) is defined as follows: for every \(F : C \rightarrow C'\), there exists \(UF : UC \rightarrow UC'\).

Let \(O\) be a fixed set. An O-graph has O as its objects; a morhpism \(D\) of \(O\)-graphs will be one with \(D_O : O \rightarrow O\) as identity. If \(A\) and \(B\) are two \(O\)-graphs, then product over \(O\) is defined as: \[A \times_O B = \{\langle f, g \rangle \mid \partial_0 f = \partial_1 g, f \in A, g \in B\}\]

The definitions \(\partial_0 \langle f, g \rangle = f, \partial_1 \langle f, g \rangle = g\) make this a graph.

A category with objects \(O\) may be described as an O-graph equipped with the two morphisms \(c : A \times_O A \rightarrow A\) and \(i : O \rightarrow A\), composition and identity, such that the following commute:

\[ \begin{xy} \xymatrix{ A \times_O A \times_O A \ar[r]^{c \times 1}\ar[d]^{1 \times c} & A \times_O A \ar[d]^c \\ A \times_O A \ar[r]^c & A } \end{xy} \begin{xy} \xymatrix{ O \times_O A \ar[r]^{i \times 1}\ar[d]^\cong & A \times_O A \ar[d]^c & A \times_O O \ar[l]_{1 \times i}\ar[d]^\cong \\ A & A & A } \end{xy} \]

A `free category` may be "generated by a graph" \(G\) having the same set of objects \(O\), and morphisms composition on edges of the graph, and this is written as \(C_G\). Given \(D : B \rightarrow C\), we get the following diagram in terms of underlying graphs:

\[ \begin{xy} \xymatrix{ G \ar[r]\ar[dr] & UB \ar@{.>}[d]|{\exists! UD} \\ & UC } \end{xy} \]

There is a bijective correspondence between categories and their underlying graphs: \[\textbf{Cat}(C_G, B) \cong \textbf{Grph}(C, UB)\]

## Quotient category

Let \(C\) be a category and \(R_{a, b}\) be a relation defined on the hom-set. Then, quotient category \(C/R\) is defined in terms of a universal functor \(Q\):

\[ \begin{xy} \xymatrix{ C \ar[r]^Q \ar[d]_H & C/R \ar@{.>}[dl]|{\exists! P} \\ D & } \end{xy} \]

If \(C = \textbf{Top}\), and \(fRf'\) is homotopy equivalence, \(C/R = \textbf{Toph}\) with objects as spaces and arrows homotopies between them. \(C/R\), as generated by a free category \(C\), is said to have generators \(G\) and relations \(R\).