Example. a cone on a diagram [kostecki2011introduction, 4.9] [tt-0029]
✍️source

The cone for a diagram

$\text{[math]}$ is $\text{[math]}$ which indeed looks like a cone with the vertex $V$.

References

Reference. An introduction to topos theory [kostecki2011introduction]

2011
Ryszard Paweł Kostecki
https://www.fuw.edu.pl/~kostecki/ittt.pdf

@article{kostecki2011introduction,
 title = {An Introduction to Topos Theory},
 author = {Kostecki, Ryszard Pawe{\l}},
 year = {2011},
 url = {https://www.fuw.edu.pl/~kostecki/ittt.pdf},
 journal = {Technial Report}
}

Context

Limits [tt-0023]

Remark. limits [leinster2016basic, ch. 5] [tt-0024]

Adjointness is about the relationships between categories. Representability is a property of set-valued functors. Limits are about what goes on inside a category.

Whenever you meet a method for taking some objects and arrows in a category and constructing a new object out of them, there is a good chance that you are looking at either a limit or a colimit.

Definition. cone [leinster2016basic, 5.1.19] [tt-0028]

Let ${\cal C}$ be a category, ${\cal J}$ a small category, and $\mathscr {D} : {\cal J} \to {\cal C}$ a diagram in ${\cal C}$ of shape ${\cal J}$.

A cone on $\mathscr {D}$ is an object $V \in {\cal C}$ (the vertex of the cone) together with a family \[ \left (V \xrightarrow {\pi _J} \mathscr {D}(J)\right )_{J \in {\cal J}} \] of arrows in ${\cal C}$ such that for all arrows $J \to J'$ in ${\cal J}$, the diagram

$\text{[math]}$ commutes.

The family of arrows are components of a natural transformation $\pi : \Delta _V \to \mathscr {D}$, i.e. from the constant functor ( which assigns the same object $V$ to any object $J_i$ in ${\cal J}$) to diagram functor $\mathscr {D}$.

For simplicity, we refer to a cone by "a cone $(V, \pi )$ on $\mathscr {D}$".

Example. a cone on a diagram [kostecki2011introduction, 4.9] [tt-0029]
✍️source

The cone for a diagram

$\text{[math]}$ is $\text{[math]}$ which indeed looks like a cone with the vertex $V$.

Definition. limit [kostecki2011introduction, 4.10] [tt-002A]

A cone $(V, \pi )$ on $\mathscr {D} : {\cal C} \to {\cal J}$ is called a limit of $\mathscr {D}$, denoted \[\lim \mathscr {D}\] if the diagram

$\text{[math]}$ commutes for every cone $(V, \pi )$ on $\mathscr {D}$.

The arrows $\pi _J$ are called the projections of the limit.

Other possible terms of limit are limiting cone, universal cone.

Example. limits [kostecki2011introduction, 4.11, example 1-4] [tt-002B]

The basic types of diagrams are actually examples of limits:

binary product [kostecki2011introduction, 2.18]: $\text{[math]}$
pullback (fiber product) [kostecki2011introduction, 2.12]: $\text{[math]}$
equalizer [leinster2016basic, 5.1.11]: $\text{[math]}$
the limit of $\mathscr {D} : \mathbf {\emptyset } \to {\cal C} $, where $\mathbf {\emptyset }$ is an empty category: $\text{[math]}$ i.e. the terminal object $\mathrm {1}$ in ${\cal C}$. In particular, for ${\cal C} = \mathbf {Set}$ we have \[\mathrm {-} \xrightarrow {!} V = \lim \mathscr {D} = \{∗\} \]

Remark. cocone, colimit [kostecki2011introduction, 4.10] [tt-002E]

A cocone and a colimit are defined by dualization, that is, by reversing the arrows in cone [leinster2016basic, 5.1.19] and limit [kostecki2011introduction, 4.10].

In another word, given $\mathscr {D}^{op} : {\cal J}^{op} \to {\cal C}^{op}$, a cocone on $\mathscr {D}$ is a cone on $\mathscr {D}^{op}$, a colimit of $\mathscr {D}$ is a limit of $\mathscr {D}^{op}$ [leinster2016basic, 5.2.1].

The arrows $\pi _J$ are called the coprojections of the colimit.

In the same say, one can and show that coequaliser, coproduct, pushout and initial object are examples of colimits.

Lemma. limits via products and equalizers [stacks2017stacks, 002N, 002P] [tt-0057]

If all products and equalizers exist, all limits exist.

Dually, if all coproducts and coequalizers exist, all colimits exist.

Definition. has (finite) limits, (finitely) complete, left exact [kostecki2011introduction, 4.10] [tt-002F]

We say that a category ${\cal C}$ has (finite) limits or is (finitely) complete if every diagram $\mathscr {D} : {\cal J} \to {\cal C}$, where ${\cal J}$ is a (finite) category, has a limit.

A category ${\cal C}$ is called left exact iff it is finitely complete.

Remark. (finitely) cocomplete, right exact [kostecki2011introduction, 4.10] [tt-003C]

Dually to Definition [tt-002F], when every diagram $\mathscr {D} : {\cal J} \to {\cal C}$, where ${\cal J}$ is a (finite) category, has a colimit, it is said that the category ${\cal C}$ has (finite) colimits or is (finitely) cocomplete.

A category is called right exact iff it is finitely cocomplete.

Lemma. (finitely) (co)complete category [kostecki2011introduction, 4.14] [tt-002J]

A category ${\cal C}$ is (finitely) complete if it has a terminal object, equalizers and (finite) products, or if it has a terminal object and (finite) pullbacks.

Dually, a category ${\cal C}$ is (finitely) cocomplete if it has an initial object, coequalizers and (finite) coproducts, or if it has an initial object and (finite) pushouts.

Lemma. (co)complete functor category [kostecki2011introduction, 4.15] [tt-002I]

If category ${\cal D}$ is complete and category ${\cal C}$ is small, then the functor category ${\cal D}^{{\cal C}}$ is complete.

Dually, if category ${\cal D}$ is cocomplete and category ${\cal C}$ is small, then the functor category ${\cal D}^{{\cal C}}$ is cocomplete.

Definition. preorder, partial order, total order [kostecki2011introduction, 1.2, example 9] [tt-002C]

Let $P$ be a set. The properties

(reflexivity) $\forall p \in P, p \leq p$
(transitivity) $\forall p, q, r \in P, p \leq q \wedge q \leq r \Rightarrow p \leq r$

define a preorder $(P, \leq )$.

A partially ordered set (called a partial order, or a poset) is defined as a preorder $(P, \leq )$ for which

(antisymmetry) $\forall p \in P, p \leq q \wedge q \leq p \Rightarrow p=q$

holds.

A total order (or a linear order) is a partial order $(P, \leq )$ for which

(comparability) $\forall p, q \in P, p \leq q \vee q \leq p$

The category $\mathbf {Preord}$ consists of objects which are preorders and of arrows which are orderpreserving functions.

The category $\mathbf {Poset}$ consists of objects which are posets and of arrows which are order-preserving functions between posets, that is, the maps $T: P \to P'$ such that \[ p \leq q \Rightarrow T(p) \leq T(q) \]

Any any preorder $(P, \leq )$ and poset $(P, \leq )$ can be considered as a category consisting of objects which are elements of a set $P$ and arrows defined by $p \to q \Longleftrightarrow p \leq q$.

Definition. directed poset [rosiak2022sheaf, def. 285] [tt-0051]

A directed poset is a poset that is inhabited (nonempty) and for which every finite subset has an upper bound. Explicitly,

(directedness) $\forall x,y \in P, \exists z \in P, x \leq z \land y \leq z$

Example. preorder, poset, directed poset [tt-0052]

An example of a preorder category which is not poset is:

$\text{[math]}$

An example of a poset category which is not a directed poset is [rosiak2022sheaf, example 3] :

$\text{[math]}$

An example that is a directed poset category but not a total order is:

$\text{[math]}$ where each pair of nodes has a common upper bound (thus satisfying directedness), but there is no path between the two nodes on the center, thus violating comparability.

A more complicated example of a directed poset category which is not a total order is [spivak2013category, example 3.4.1.3]:

$\text{[math]}$

One can see immediately that this is a preorder because length=0 paths give reflexivity and concatenation of paths gives transitivity. To see that it is a partial order we only note that there are no loops.

To see that it is a poset, we note that every pair of nodes from one side or both sides has the central node as an upper bound, thus satisfying directedness.

But this partial order is not a total order because there is no path (in either direction) between some nodes, thus violating comparability.

Definition. inverse limit, projective limit [kostecki2011introduction, 4.11] [tt-002D]

Let ${\cal J}$ be a directed poset and $\mathscr {F} : {\cal J} \to {\cal C}$ be a contravariant functor. The limit of $\mathscr {F}$ is called an inverse limit or projective limit, and is denoted $\lim \limits _{\leftarrow {\cal J}} \mathscr {F}$ or simply $\lim \limits _{\longleftarrow } \mathscr {F}$.

Definition. direct limit, inductive limit [kostecki2011introduction, 4.12] [tt-002G]

Let ${\cal J}$ be a directed poset and $\mathscr {F}: {\cal J} \to {\cal C}$ be a contravariant functor. The colimit of $\mathscr {F}$ is called a direct limit (some called directed limit) or inductive limit, and is denoted $\lim \limits _{\to {\cal J}} \mathscr {F}$, or simply $\lim \limits _{\longrightarrow } \mathscr {F}$.

This is dual to inverse limit.

Definition. preserves (all) (co)limits, left/right exact [kostecki2011introduction, 4.13] [tt-002H]

A functor $\mathscr {F}: {\cal C} \to {\cal D}$ preserves (all) limits and is called left exact iff it sends all limits in ${\cal C}$ into limits in ${\cal D}$.

Dually, a functor $\mathscr {F}: {\cal C} \to {\cal D}$ preserves (all) colimits and is called right exact iff it sends all colimits in ${\cal C}$ into colimits in ${\cal D}$.

Remark. directions in (co)limits [tt-004Y]

	Limit	Colimit
diagram	$\text{[math]}$	$\text{[math]}$
arrows through the vertex	into the diagram	out of the diagram
on (co)shape P	pullback	pushout
categories have finite ...	left exact	right exact
functors preserve all ...	left exact	right exact
on directed poset	inverse/projective limit $\lim \limits _{\longleftarrow } \mathscr {F}$	direct/inductive limit $\lim \limits _{\longrightarrow } \mathscr {F}$

One can see from the table that, in general, limits have the direction "back" "into" (where "back", "left", "inverse" are directional consistent), and colimits have the opposite: "forward" "out of".

This might help to memorize the directions in these concepts without disorientation.