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

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.

Context

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]
✍️source

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.

Backlinks

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

Diagramatically,

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

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.

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]:
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 \[\mathrm {-} \xrightarrow {!} V = \lim \mathscr {D} = \{∗\} \]

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.

definition. subobject classifier [kostecki2011introduction, 6.4] [tt-004I]

A subobject classifier is an object \(\Omega \) in \({\cal C}\), together with an arrow \(\top : \mathbf {1} \to \Omega \), called the true arrow, such that for each monic arrow \(m: Y \mapsto X\) there is a unique arrow \(\chi : X \to \Omega \), called the characteristic arrow of \(m\) (or of \(Y\)), such that the diagram is a pullback, where \(!\) is the unique functor.

\(\Omega \) is also called a generalized truth-value object.

The arrow \((! \mathbin {\bullet } T) : Y \stackrel {!}{\to } \mathbf {1} \rightarrowtail \Omega \) is often denoted as \(\top _Y: Y \to \Omega \).

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}
}