Morphisms and products

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 11 August 2012

Morphisms

A groupoid is a category such that all morphisms are isomorphisms.

An isomorphism is a morphism $f : X \to Y$ such that there exists a morphism $g : Y \to X$ such that $f \circ g = {id}_{Y}$ and $g \circ f = {id}_{X}$ .

Let $f : X \to Y$ be a morphism.

A section, or right inverse, of $f$ is a morphism $g : Y \to X$ such that $f \circ g = {id}_{Y}$ .
A retraction, or left inverse, of $f$ is a morphism $g : Y \to X$ such that $g \circ f = {id}_{X}$ .
A monomorphism is a morphism $f : X \to Y$ with the left cancellation property, i.e. a morphism $f : X \to Y$ such that if $g_{1} : Z \to X$ $g_{2} : Z \to X$ and $f \circ g_{1} = f \circ g_{2}$ then $g_{1} = g_{2}$ .
A epimorphism is a morphism $f : X \to Y$ with the right cancellation property, i.e. a morphism $f : X \to Y$ such that if $g_{1} : Y \to Z$ $g_{2} : Y \to Z$ and $g_{1} \circ f = g_{2} \circ f$ then $g_{1} = g_{2}$ .

Let $X$ be an object in $𝒞$ .

A subobject of $X$ is an isomorphism class of a monomorphism $f : Y \to X$ .
A quotient of $X$ is an isomorphism class of an epimorphism $f : X \to Q$ .

Limits, initial and terminal objects, sums and products, kernels and cokernels, fiber products and coproducts

The fiber product, or pullback, of

is $X_{0} \times_{Y} X_{1}$ given by

i.e., it is given by

such that if

then there exists a unique $Z ⇢ X_{0} \times_{Y} X_{1}$ such that

commutes.

The fiber coproduct, or pushout, of

is

such that if

then there exists a unique $Y_{0} ⊔_{X} Y_{1} ⇢ Z$ such that

commutes.
The product of $X_{0}$ and $X_{1}$ is $X_{0} \times X_{1}$ given by

(In the category Set this is product.)
The coproduct of $Y_{0}$ and $Y_{1}$ is $Y_{0} ⊔ Y_{1}$ given by

(In the category Set this is disjoint union.)
The kernel of $X_{0} ⇉_{g}^{f} X_{1}$ is $K$ given by
The cokernel of $X_{0} ⇉_{g}^{f} X_{1}$ is $L$ given by

HW: Show that, in a small category, $X_{0} \times_{Y} X_{1} = {(x, y) \in X_{0} \times X_{1} | f (x) = g (y)} and Y_{0} ⊔_{X} Y_{1} = \frac{Y_{0} ⊔ Y_{1}}{⟨ f (x) = g (x) ⟩} .$

Let $𝒞$ be an additive category and let $f : X \to Y$ be a morphism.

The kernel of $f$ is the fiber product of
The cokernel of $f$ is the fiber coproduct of

An inductive system indexed by $I$ is a functor $α : I \to 𝒞$ .
An projective system indexed by $I$ is a functor $α : I^{op} \to 𝒞$ .
The inductive limit of $α : I \to 𝒞$ is $\lim_{\to} α$ given by

for $i, j \in I$ and $s : i \to j$ .
The projective limit of $β : I^{op} \to 𝒞$ is $\lim_{\leftarrow} β$ given by

for $i, j \in I$ and $s : i \to j$ .

Fiber products and coproducts are inductive and projective limits corresponding to the category

I = • \leftarrow • \to •

Products and sums are inductive and projective limits corresponding to the category

I = • • • \dots

(the discrete category with set

I

Cokernels and kernels are inductive and projective limits corresponding to the category

I = • ⇉ •

Notes and References

These notes are distilled from [KS, Chapt. 1-2].

This page http://ncatlab.org/nlab/show/HowTo contains the following example:

This page http://golem.ph.utexas.edu/~distler/blog/archives/001594.html contains the following example:

Pairing of states as an (n-1) functor $C$ $C$ $C$ $C$ $V_{x}$ $V_{y}$ $V_{x'}$ $V_{y'}$ $e_{1} (x)$ $⇓$ $e_{1} (y)$ ${\bar{e}}_{2} (x')$ $⇓$ ${\bar{e}}_{2} (y')$ $tra (γ_{1})$ $⇓ tra (Σ)$ $tra (γ_{2})$ $X$

References

[KS] M. Kashiwara and P. Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften 332 Springer-Verlag, Berlin, 2006. x+497 pp. ISBN: 978-3-540-27949-5; 3-540-27949-0 MR2182076.

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