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

The basic types of diagrams are actually examples of limits:

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

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 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]

The cone for a diagram is 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 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]
✍️source

The basic types of diagrams are actually examples of limits:

binary product [kostecki2011introduction, 2.18]:
pullback (fiber product) [kostecki2011introduction, 2.12]:
equalizer [leinster2016basic, 5.1.11]:
the limit of \(\mathscr {D} : \mathbf {\emptyset } \to {\cal C} \), where \(\mathbf {\emptyset }\) is an empty category: 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:

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

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

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
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.

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}
}

basic types of diagrams [tt-000P]

By "basic types of diagrams", we mean some basic structures inside a category.

definition. diagram, shape [leinster2016basic, 5.1.18] [tt-0025]

Let \({\cal C}\) be a category and \({\cal J}\) a small category. A functor \(\mathscr {D} : {\cal J} \to {\cal C}\) is called a diagram in \({\cal C}\) of shape \({\cal J}\).

\({\cal J}\) is also called the indexing category of the diagram, and we say that \(\mathscr {D}\) is a diagram indexed by \({\cal J}\) [rosiak2022sheaf, example 35]. \(J\) is also called the template.

definition. shape E [leinster2016basic, 5.14] [tt-000O]

The diagram is called a diagram of shape E.

For simplicity, we refer to a diagram of shape E by "a shape \(E(f, g)\)".

"E" in "shape E" stands for "equal", and the reason will unfold in the definition of equalizer.

definition. fork [leinster2016basic, 5.4] [tt-000R]

A fork over a shape \(E(f, g)\) is the diagram that makes the diagram commute.

For simplicity, we refer to a fork by "a fork \((E, \iota )\) (over the shape \(E(f, g)\))".

convention. grey arrow [tt-000S]

We use grey arrows to represent the composition arrow in a fork. This convention is not from literatures and is subject to change.

definition. equalizer [leinster2016basic, 5.1.11] [tt-000Q]

An equalizer of a shape \(E(f, g)\) is a fork \((E, \iota )\) over it, such that, for any \((\mathrm {-}, d)\) over the fork, the diagram commutes (i.e. any arrow \(d : \mathrm {-} \to X\) must uniquely factor through \(E\)).

For simplicity, we refer to the equalizer of a shape \(E(f, g)\) as \(\operatorname {Eq}(f, g)\), and \(\iota \) is the canonical inclusion.

We say that a category \({\cal C}\) has equalizers iff every shape E in \({\cal C}\) has an equalizer.

remark. equalizing set [kostecki2011introduction, eq. 32] [tt-000U]

Equalizer in a category is a generalisation of a subset which consists of elements of a given set such that two given functions are equal on them, formally:

For any two arrows \(f, g: X \to Y\), their equalizing set \(E \subseteq X\) is defined as \[ E:=\{e \mid e \in X \wedge f(e)=g(e)\} \]

definition. shape P [leinster2016basic, 5.14] [tt-000W]

The diagram is called a diagram of shape P.

For simplicity, we refer to a diagram of shape P by "a shape \(P(f, g)\)".

"P" in "shape P" may stand for "product/projection/pullback", and the reason will unfold in the definition of pullback.

definition. pullback (fiber product) [kostecki2011introduction, 2.12] [tt-000V]

A pullback of a shape \(P(f, g)\) is an object \(X \times _O Y\) in \({\cal C}\) together with arrows \(p_X\) and \(p_Y\), called projections, such that, for any object \(\mathrm {-}\) and arrows \(h\) and \(k\), the diagram commutes.

We say that a category \({\cal C}\) has pullbacks iff every shape \(P(f, g)\) in \({\cal C}\) has a pullback in \({\cal C}\).

A pullback is also called a fiber product.

The square is called the pullback square of \(f\) and \(g\). The object \(X \times _O Y\) in \({\cal C}\) is called the fiber product object.

lemma. pasting pullbacks [spivak2013category, 2.5.1.17] [tt-0056]

Pullbacks can be pasted together, i.e. for diagram given that the right-hand square is a pullback, the left-hand square is a pullback if and only if the outer rectangle is a pullback.

definition. shape T [leinster2016basic, 5.14] [tt-0026]

The diagram is called a diagram of shape T.

For simplicity, we refer to a diagram of shape T by "a shape \(T(X, Y)\)" where \(X\) and \(Y\) are the 2 objects.

"T" in "shape T" stands for "two". Shape T is useful in the definition of binary product [kostecki2011introduction, 2.18].

definition. binary product [kostecki2011introduction, 2.18] [tt-000Y]

A binary product of objects \(X\) and \(Y\) is an object \(X \times Y\) in \({\cal C}\) together with arrows \(p_X\) and \(p_Y\), called projections, such that, for any object \(\mathrm {-}\) and arrows \(h\) and \(k\), the diagram

commutes.

We say that a category \({\cal C}\) has binary products iff every pair \(X, Y\) in \({\cal C}\) has a binary product \(X \times Y\) in \({\cal C}\).

When there is no confusion, we simply call bianry products products.

definition. coshape, coequalizer, pushout (fiber coproduct), binary coproduct [kostecki2011introduction, 2.14, 2.16, 2.19] [tt-000X]

coshape, coequalizer, pushout (fiber coproduct), binary coproduct can be defined by reversing all arrows in the definitions of shape, equalizer, pullback (fiber product), binary product respectively.

The pushout equivalent of the fiber product object in pullback is the fiber coproduct object, denoted \(X +_{O} Y\), and the pushout equivalent of projections in pullback are injections, denoted \(i_X\) and \(i_Y\), respectively. The unique arrow of a pushout is denoted \([f, g]\).

The binary coproduct equivalent of the binary product object in binary product is the binary coproduct object, denoted \(X + Y\), and the binary coproduct equivalent of projections in binary product are injections, denoted \(i_X\) and \(i_Y\), respectively. The unique arrow of a binary coproduct is denoted \([f, g]\).

Diagramatically,

Coshapes:
- \(T\) =
- \(E\) =
- \(P\) =
coequalizer:
pushout (fiber coproduct):
binary coproduct:

lemma. monic and pullback [leinster2016basic, 5.1.32] [tt-003J]

An arrow \(X \xrightarrow {f} Y\) is monic iff the square is a pullback.

The significance of this lemma is that whenever we prove a result about limits, a result about monics will follow.

lemma. epic and pushout [leinster2016basic, sec. 5.2] [tt-003M]

An arrow \(X \xrightarrow {f} Y\) is epic iff the square is a pushout.

This is dual to lemma [tt-003J].

definition. n-fold (co)products [kostecki2011introduction, 2.22] [tt-0011]

In any category with binary products the objects \(X \times (Y \times Z)\) and \((X \times Y) \times Z\) are isomorphic. In any category with binary coproducts the objects \(X+(Y+Z)\) and \((X+Y)+Z\) are isomorphic.

This allows to consider \(n\)-fold products \(X_1 \times \dots \times X_n\) and \(n\)-fold coproducts \(X_1 + \dots + X_n\) of objects of a given category.

definition. have finite (co)products [kostecki2011introduction, 2.23] [tt-0012]

A category which has n-fold (co)products for any \(n \in \mathbb N\) is said to have finite (co)products.