Universal construction defines objects in terms of their relations with other objects.

1 Terminal object & initial object

• A terminal object is an object $$1$$ that for every object $$A$$ there exists a unique morphism $$e_A: A \to 1$$ to it.
• An initial object is an object $$0$$ that for every object $$B$$ there exists a unique morphism $$i_B: 0 \to B$$ from it.

\begin{aligned} \begin{array}{rlcll} \text{terminal object:} & 1 & \equiv & \forall A. & \exists! e_A: A \to 1. \\ \text{initial object:} & 0 & \equiv & \forall B. & \exists! i_A: 0 \to B. \end{array} \end{aligned}

• A weakly terminal object is an object that for every object there exists at least one morphism to it.
• A weakly initial object is an object that for every object there exists at least one morphism from it.
• A subterminal object is an object that for every object there exists at most one morphism to it.
• A subinitial object is an object that for every object there exists at most one morphism from it.
• A strictly terminal object is a terminal object $$1$$ that every morphism from it is a isomorphism.
• A strictly initial object is an initial object $$0$$ that every morphism to it is a isomorphism.
• A zero object $$0$$ is both a terminal object and an initial object.
• A zero morphism $$0_{A, B}: A \to 0 \to B$$ is the unique morphism that factors through the zero object $$0$$.
• A pointed category is a category with a zero object.
• A generalized element of $$A$$ is a morphism $$a: X \to A$$.
• A global element of $$A$$ is a morphism $$a: 1 \to A$$.

2 Universal morphism

\begin{aligned} \begin{array}{ccccc} F: & \cC & \toot & \cD & :G \\ & A && B & \\ & f && g & \end{array} \end{aligned}

• A universal morphism from $$F$$ to $$B$$ is a terminal object $$(T, FT \xto{\epsilon} B)$$ in the comma category $$(F: \cC \to \cD) \comma (B: \cOne \to \cD)$$.
• A universal morphism from $$A$$ to $$G$$ is an initial object $$(I, A \xto{\eta} GI)$$ in the comma category $$(A: \cOne \to \cC) \comma (G: \cD \to \cC)$$.

\begin{aligned} \begin{array}{rlclll} \text{universal morphism from F to B:} & (T, FT \xto{\epsilon} B) & \equiv & \forall g: FA \to B. & \exists! \hat{g}: A \to T. & g = \epsilon \compL F\hat{g}. \\ \text{universal morphism from A to G:} & (I, A \xto{\eta} GI) & \equiv & \forall f: A \to GB. & \exists! \hat{f}: I \to B. & f = G\hat{f} \compL \eta. \end{array} \end{aligned}

• A right adjoint functor of $$F$$ is a universal morphism $$(G, \epsilon: FG \nat \id_\cD)$$ from $$F^\cD: \cC^\cD \to \cD^\cD$$ to $$\id_\cD$$.
• A left adjoint functor of $$G$$ is a universal morphism $$(F, \eta: \id_\cC \nat GF)$$ from $$\id_\cC$$ to $$G^\cC: \cD^\cC \to \cC^\cC$$.
• For each $$\cD$$-object $$B$$, there exists a universal morphism $$(GB, \epsilon_B: FGB \to B)$$ from $$F$$ to $$B$$.
• For each $$\cC$$-object $$A$$, there exists a universal morphism $$(FA, \eta_A: A \to GFA)$$ from $$A$$ to $$G$$.

$G(B \xto{g} B') := \widehat{g \compL \epsilon_B}$

$F(A \xto{f} A') := \widehat{\eta_A' \compL f}$

3.2 Hom-set isomorphism

$\alpha: \Hom_\cD(FA, B) \iso \Hom_\cC(A, GB) :\alpha\inv$

\begin{aligned} \begin{array}{cccc} \alpha\inv_{GB, B}: & \Hom_\cC(GB, GB) & \iso & \Hom_\cD(FGB, B) \\ & \id_{GB} & \mapsto & \epsilon_B \\ \alpha_{A, FA}: & \Hom_\cD(FA, FA) & \iso & \Hom_\cC(A, GFA) \\ & \id_{FA} & \mapsto & \eta_A \end{array} \end{aligned}

\begin{aligned} \begin{array}{rll} \text{isomorphism:} & \id_\cC = GF & FG = \id_\cD \\ \text{equivalence:} & \id_\cC \iso GF & FG \iso \id_\cD \\ \text{adjunction:} & \id_\cC \nat GF & FG \nat \id_\cD \end{array} \end{aligned}

$\cC \adjto{F}{G} \cD$

An adjunction $$F \adj G$$ consists of a pair of adjoint functors $$F: \cC \toot \cD: G$$ with

• counit $$\epsilon: FG \nat \id_\cD$$
• unit $$\eta: \id_\cC \nat GF$$

such that

• $$(\epsilon F) \compL (F \eta) = \id_F$$
• $$(G \epsilon) \compL (\eta G) = \id_G$$

3.4 Galois connection

$F: (A, \leq_A) \adj (B, \leq_B) :G$

\begin{aligned} Fa := \min \set{b \in B \mid a \leq_A Gb} \\ Gb := \max \set{a \in A \mid Fa \leq_B b} \end{aligned}

3.4.1 Multiplication and division

$(\R, \leq) \adjto{- / c}{- \times c} (\R, \leq)$

$a / c \leq b \equiv a \leq b \times c$

• $$\eta_a: a \leq a / c * c = \id_a$$
• $$\epsilon_a: a * c / c \leq a = \id_a$$

3.4.2 Union and intersection

$(PA, \subseteq) \adjto{- \cap X}{\overline{X} \cup -} (PA, \subseteq)$

$Z \cap X \subseteq Y \equiv Z \subseteq \overline{X} \cup Y$

• $$\eta_A: A \subseteq \overline{X} \cup (A \cap X)$$
• $$\epsilon_A: (\overline{X} \cup A) \cap X \subseteq A$$

3.4.3 Floor and ceiling

$(\R, \leq) \adjto{\ceil{-}}{} (\Z, \leq) \adjto{}{\floor{-}} (\R, \leq)$

$\ceil{x} \leq_\Z n \equiv x \leq_\R n$

$n \leq_\R x \equiv n \leq_\Z \floor{x}$

• $$\eta_x: x \leq_\R \ceil{x}$$
• $$\epsilon_n: \ceil{n} \leq_\Z n = \id_n$$
• $$\eta_n: n \leq_\Z \floor{n} = \id_n$$
• $$\epsilon_x: \floor{x} \leq_\R x$$

3.4.4 Maximum and minimum

$\cOne \adjto{\min A}{} (A, \leq) \adjto{}{\max A} \cOne$

• $$\eta_* = \id_*$$
• $$\epsilon_a: \min A \leq a$$
• $$\eta_a: a \leq \max A$$
• $$\epsilon_* = \id_*$$

$(PA, \subseteq) \adjto{\max}{A_{\leq -}} (A, \leq) \adjto{A_{\geq -}}{\min} (PA, \supseteq)$

$\max A \leq b \equiv A \subseteq A_{\leq b}$

$A_{\geq a} \supseteq B \equiv a \leq \min B$

• $$\eta_A: A \subseteq A_{\leq \max A}$$
• $$\epsilon_b: \max A_{\leq b} \leq b = \id_b$$
• $$\eta_a: a \leq \min A_{\geq a} = \id_a$$
• $$\epsilon_B: A_{\geq \min B} \supseteq B$$

$(PA, \subseteq) \adjto{\min}{A_{\geq -}} (A, \geq) \adjto{A_{\leq -}}{\max} (PA, \supseteq)$

$\min A \geq b \equiv A \subseteq A_{\geq b}$

$A_{\leq a} \supseteq B \equiv a \geq \max B$

• $$\eta_A: A \subseteq A_{\geq \min A}$$
• $$\epsilon_b: \min A_{\geq b} \geq b = \id_b$$
• $$\eta_a: a \geq \max A_{\leq a} = \id_a$$
• $$\epsilon_B: A_{\leq \max B} \supseteq B$$

$\cC \adjto{F\text{ree}}{U\text{nderlying}} \cD$

4 Kan

• $$F: \cA \to \cC$$
• $$G: \cA \to \cB$$
• $$H: \cB \to \cC$$
• $$\cC^G: \cC^\cB \to \cC^\cA$$ precomposition
• $$H^\cA: \cB^\cA \to \cC^\cA$$ postcomposition
• $$\Lan_G, \Ran_G: \cC^\cA \to \cC^\cB$$ Kan extension
• $$\Lift_H, \Rift_H: \cC^\cA \to \cB^\cA$$ Kan lift

4.1 Kan extension

• A right Kan extension of $$F$$ through $$G$$ is a universal morphism $$(\Ran_G F: \cB \to \cC, \epsilon_F: \Ran_G F G \nat F)$$ from $$\cC^G$$ to $$F$$.
• A left Kan extension of $$F$$ through $$G$$ is a universal morphism $$(\Lan_G F: \cB \to \cC, \eta_F: F \nat \Lan_G F G)$$ from $$F$$ to $$\cC^G$$.

$\Lan_G \adj \cC^G \adj \Ran_G$

4.2 Kan lift

• A right Kan lift of $$F$$ through $$H$$ is a universal morphism $$(\Rift_H F: \cA \to \cB, \epsilon_F: H \Rift_H F \nat F)$$ from $$H^\cA$$ to $$F$$.
• A left Kan lift of $$F$$ through $$H$$ is a universal morphism $$(\Lift_H F: \cA \to \cB, \eta_F: F \nat H \Lift_H F)$$ from $$F$$ to $$H^\cA$$.

$\Lift_H \adj H^\cA \adj \Rift_H$

5 Limit & colimit

A diagram of type $$\cI$$ in a category $$\cC$$ is a functor

\begin{aligned} D: \cI & \to \cC \\ i & \mapsto D_i \\ ij & \mapsto D_{ij} \end{aligned}

$$\cI$$ is called the index category.

Given the diagonal functor $$\Delta: \cC \to \cC^\cI$$ and a diagram $$D: \cI \to \cC$$:

• A limit is a universal morphism $$(\lim D, \epsilon_D: \Delta \lim D \nat D)$$ from $$\Delta$$ to $$D$$.
• A colimit is a universal morphism $$(\colim D, \eta_D: D \nat \Delta \colim D)$$ from $$D$$ to $$\Delta$$.

\begin{aligned} \begin{array}{rlclll} \text{limit:} & \epsilon_D: \Delta \lim D \nat D & \equiv & \forall c: \Delta C \nat D. & \exists! \hat{c}: C \to \lim D. & c = \epsilon_D (\Delta \hat{c}). \\ \text{colimit:} & \eta_D: D \nat \Delta \colim D & \equiv & \forall c: D \nat \Delta C. & \exists! \hat{c}: \colim D \to C. & c = (\Delta \hat{c}) \eta_D. \end{array} \end{aligned}

$\colim \adj \Delta \adj \lim$

• A (finitely) (co)complete category has all (finite) (co)limits.
• A (finitely) bicomplete category is (finitely) complete and (finitely) cocomplete.

5.1 Cone & cocone

A cone $$(C, c)$$ over a diagram $$D: \cI \to \cC$$ consists of

• a $$\cC$$-object $$C$$
• a collection of $$\cC$$-morphisms from $$C$$ indexed by $$\cI$$-objects $$c_i: C \to D_i$$

such that for all $$\cI$$-morphisms $$ij: i \to j$$, $$c_j = D_{ij} \compL c_i$$.

A cone morphism $$f: (C, c) \to (C', c')$$ is a $$\cC$$-morphism $$f: C \to C'$$ that for all $$\cI$$-objects $$i$$, $$c_i = c'_i \compL f$$.

A cone category $$\cCone(D)$$ consists of all cones over a diagram $$D$$ and all cone morphisms.

A limit $$\lim D$$ of a diagram $$D$$ is a terminal object in $$\cCone(D)$$.

• A limit is a cone: (a $$\cC$$-object $$\lim D$$, a collection of $$\cC$$-morphisms $$\epsilon_i: \lim D \to D_i$$)
• A limit is a terminal object: for all cones $$(C, c)$$, there exists a unique cone morphism $$f: (C, c) \to (\lim D, \epsilon)$$ such that $$c_i = \epsilon_i \compL f$$.

The concepts of cocone, cocone morphism, cocone category and colimit are defined dually.

• cone: $$c: \Delta C \nat D$$, cocone: $$c: D \nat \Delta C$$
• cone/cocone morphism: $$\Delta f: \Delta C \nat \Delta C'$$
• cone category: $$\Delta \comma D$$, cocone category: $$D \comma \Delta$$
• limit: $$\epsilon_D: \Delta \lim D \nat D$$, colimit: $$\eta_D: D \nat \Delta \colim D$$

5.2 Preservation

For a diagram $$D: \cI \to \cC$$ and a functor $$F: \cC \to \cD$$,

• $$F$$ preserves a limit $$(\lim D, \epsilon_D)$$ in $$\cC$$ if $$(F \lim D, F \epsilon_D)$$ is a limit in $$\cD$$
• $$F$$ preserves a colimit $$(\colim D, \eta_D)$$ in $$\cC$$ if $$(F \colim D, F \eta_D)$$ is a colimit in $$\cD$$

A (finitely) (co)continuous functor preserves all (finite) (co)limits.

$$F$$ is a left adjoint functor $$\equiv$$ $$F$$ preserves colimits. $$G$$ is a right adjoint functor $$\equiv$$ $$G$$ preserves limits.

Limits commute with limits; colimits commute with colimits.

$$\Hom_\cC$$ preserves limits.

$\Hom_\cC(C, \lim D) \iso \lim \Hom_\cC(C, -) D$

$\Hom_\cC(\colim D, C) \iso \lim \Hom_\cC(-, C) D$

$\Hom_\cC(C, A \times B) \iso \Hom_\cC(C, A) \times \Hom_\cC(C, B)$

$\Hom_\cC(A + B, C) \iso \Hom_\cC(A, C) \times \Hom_\cC(B, C)$

6 Examples

Tarski’s high school algebra (in a bicartesian closed category):

1. $$A + B \iso B + A$$
2. $$(A + B) + C \iso A + (B + C)$$
3. $$A \times 1 \iso A$$
4. $$A \times B \iso B \times A$$
5. $$(A \times B) \times C \iso A \times (B \times C)$$
6. $$A \times B + A \times C \iso A \times (B + C)$$
7. $$1^A \iso 1$$
8. $$A^1 \iso A$$
9. $$A^{B + C} \iso A^B \times A^C$$
10. $$(A \times B)^C \iso A^C \times B^C$$
11. $$(A^B)^C \iso A^{B \times C}$$

6.1 Set

 terminal object singleton $$\set{*}$$ initial object empty set $$\varnothing$$ product cartesian product $$A \times B$$ coproduct disjoint union $$A + B$$ equalizer $$\Eq(f, g: A \to B) = \set{a \in A \mid f(a) = g(a)}$$ coequalizer $$\Coeq(f, g: A \to B) = B / \sim$$ where $$b_1 \sim b_2$$ if $$\exists a \in A. f(a) = b_1 \land g(a) = b_2$$ pullback $$\Pull(f: A \to C, g: B \to C) = \set{(a, b) \in A \times B \mid f(a) = g(b)}$$ pushout $$\Push(f: C \to A, g: C \to B) = (A + B) / \sim$$ where $$a \sim b$$ if $$\exists c \in C. f(c) = a \land g(c) = b$$ fiber inverse image $$\Pull(f: A \to B, b: 1 \to B) = \set{a \in A \mid f(a) = b}$$ monomorphism injective function epimorphism surjective function subobject subset subobject classifier indicator function power object power set exponential object the set of functions and function evaluation

6.2 Cat

 terminal object $$\cOne$$ initial object $$\cZero$$ product product category $$\cC \times \cD$$ exponential object functor category $$[\cC, \cD]$$

6.3 Poset

$$(A, \leq)$$

 terminal object greatest element / top $$\top$$ initial object least element / bottom $$\bot$$ product greatest lower bound / meet $$a \meet b$$ coproduct least upper bound / join $$a \join b$$ equalizer $$\id_a: a \leq a$$ coequalizer $$\id_b: b \leq b$$ pullback $$p_1: a \meet b \leq a$$, $$p_2: a \meet b \leq b$$ pushout $$i_1: a \leq a \join b$$, $$i_2: a \leq a \join b$$ exponential object $$a \ihom b := \bigjoin\set{c \mid c \lcon a \leq b}$$ ($$c \lcon a \leq b \equiv c \leq a \ihom b$$)
• A meet-semilattice is a poset with all binary meets $$a \meet b$$.
• A join-semilattice is a poset with all binary joins $$a \join b$$.
• A lattice is a meet-semilattice and a join-semilattice.
• A bounded poset is a poset with a terminal object $$\top$$ and/or an initial object $$\bot$$.
• A bounded meet-semilattice is a finitely complete poset.
• A bounded join-semilattice is a finitely cocomplete poset.
• A bounded lattice is a finitely bicomplete poset.
• A Heyting algebra is a cartesian closed bounded lattice.
• A Boolean algebra is a Heyting algebra satisfying the law of excluded middle.

A Heyting algebra is finitely distributive because it is finitely bicartesian and closed.

• A inflattice is a complete poset.
• A suplattice is a cocomplete poset.
• A complete lattice is a bicomplete poset.

By the adjoint functor theorem for posets, all joins $$\equiv$$ all meets, i.e., inflattice = suplattice = complete lattice.

A inflattice/suplattice homomorphism is a monotone preserving meets/joins.

A complete Heyting algebra is a cartesian closed complete lattice.

6.3.1$$(\N, \leq)$$

 terminal object $$-$$ initial object $$0$$ $$0 \leq a$$ product $$\min$$ $$c \leq a, c \leq b \equiv c \leq \min(a, b)$$ coproduct $$\max$$ $$a \leq c, b \leq c \equiv \max(a, b) \leq c$$ exponential object $$-$$

6.3.2$$(\N, \mid)$$

 terminal object $$0$$ $$a \mid 0$$ initial object $$1$$ $$1 \mid a$$ product $$\gcd$$ $$c \mid a, c \mid b \equiv c \mid \gcd(a, b)$$ copoduct $$\lcm$$ $$a \mid c, b \mid c \equiv \lcm(a, b) \mid c$$ exponential object $$-$$

6.3.3$$(\set{\lfal, \ltru}, \lfal \lproves \ltru)$$

 terminal object $$\ltru$$ initial object $$\lfal$$ product $$\lcon$$ $$C \lproves A, C \lproves B \equiv C \lproves A \lcon B$$ copoduct $$\ldis$$ $$A \lproves C, B \lproves C \equiv A \ldis B \lproves C$$ exponential object $$\limp$$ $$C \lcon A \lproves B \equiv C \lproves (A \limp B)$$

6.3.4$$([0, \infty], \geq)$$

 terminal object $$0$$ $$a \geq 0$$ initial object $$\infty$$ $$\infty \geq a$$ product $$\max$$ $$c \geq a, c \geq b \equiv c \geq \max(a, b)$$ copoduct $$\min$$ $$a \geq c, b \geq c \equiv \min(a, b) \geq c$$ exponential object $$\begin{cases}0 & a \geq b \\ b & \text{otherwise}\end{cases}$$ $$\max(c, a) \geq b \equiv c \geq \begin{cases}0 & a \geq b \\ b & \text{otherwise}\end{cases}$$

6.4 Met

metric spaces $$(A, d_A: A \times A \to [0, \infty])$$ and metric functions $$d_A(a, a') \geq d_B(f(a), f(a'))$$.

 terminal object singleton $$(\set{*}, d)$$ initial object empty set $$(\varnothing, d)$$ product product metric $$d_{A \times B}((a, b), (a', b')) := \max(d_A(a, a'), d_B(b, b'))$$ coproduct disjoint metric $$d_{A + B}(a, a') := d_A(a, a')$$, $$d_{A + B}(b, b') := d_B(b, b')$$, $$d_{A + B}(a, b) = d_{A + B}(b, a) := \infty$$