**math.ct**34

abstract algebra - Coproducts exist in the category of groups - Mathematics Stack Exchange

october 2018 by quamash

Let A,B be groups, then C (with homomorphisms ia:A→C, ib:B→C) is the coproduct if given any C′ with group homomorphisms a:A→C′, b:B→C′ there exists a unique map i:C→C′ such that a=i∘ia and b=i∘ib

. That's just taking the general definition of a coproduct and writing it out with "groups" instead of "objects" and "group homomorphisms" instead of "morphisms" so we haven't done anything yet.

Now I claim that if we let C

be the free product of groups A⋆B (which always exists) and ia,ib the inclusion maps then this is the coproduct.

math.CT
. That's just taking the general definition of a coproduct and writing it out with "groups" instead of "objects" and "group homomorphisms" instead of "morphisms" so we haven't done anything yet.

Now I claim that if we let C

be the free product of groups A⋆B (which always exists) and ia,ib the inclusion maps then this is the coproduct.

october 2018 by quamash

Dr Julia Goedecke - Teaching

august 2018 by quamash

Cambridge Maths Part III Category Theory lecture course

In Michaelmas 2013 I lectured the Part III course on Category Theory, as I also did in Michaelmas 2011 and 2012. Material is still here for interested people.

Lecture Notes

2013 example sheets:

Example Sheet 1

Example Sheet 2

Example Sheet 3

Example Sheet 4

Further examples:

Examples of categories and functors

Examples of adjunctions

Examples of coseparating sets and separating families

Notes on Adjunctions, Monads and Lawvere Theories by Filip Bár

Example sheets from previous years: on archive page.

Video Solutions

Video solutions to some example sheet questions.

You should only look at these once you've tried the question! Some data will be saved when you watch the videos, such as where you pause, what answer you clicked, etc. You can use the tick boxes on the list of videos to indicated if you have "completed" a question; I will also be able to see this information.

math.CT
math.AT
prelims
In Michaelmas 2013 I lectured the Part III course on Category Theory, as I also did in Michaelmas 2011 and 2012. Material is still here for interested people.

Lecture Notes

2013 example sheets:

Example Sheet 1

Example Sheet 2

Example Sheet 3

Example Sheet 4

Further examples:

Examples of categories and functors

Examples of adjunctions

Examples of coseparating sets and separating families

Notes on Adjunctions, Monads and Lawvere Theories by Filip Bár

Example sheets from previous years: on archive page.

Video Solutions

Video solutions to some example sheet questions.

You should only look at these once you've tried the question! Some data will be saved when you watch the videos, such as where you pause, what answer you clicked, etc. You can use the tick boxes on the list of videos to indicated if you have "completed" a question; I will also be able to see this information.

august 2018 by quamash

at.algebraic topology - Is Mac Lane still the best place to learn category theory? - MathOverflow

august 2018 by quamash

I doubt that someone could learn higher category theory (and more in general higher dimensional algebra) without first studying a little of category theory, mostly because the definition given in such context use a lot of category theoretic machinery. About the textbook reference: MacLane's "Category theory for working mathematicians" may be a little outdated but I think it is still one of the most complete book of basic category theory second just to Borceux's books. Anyway there isn't a best book to learn basic category theory, any person could find a book better than another one, so I suggest you to take a look a some of these books, then choose which one is the best for you:

S. MacLane: Category theory for working mathematicians (I've already said a lot about this)

S. Awodey: Category theory (Peculiar because it has very low prerequisites and it's rich of examples too)

J. Adamek,H. Herrlich, G. Strecker: Abstract and concrete category theory (freely avaible at at this site "http://katmat.math.uni-bremen.de/acc/acc.pdf", maybe the book with the greatest number of examples from topology and algebra)

After you have read one of these book, you could also use Borceux's books and read some more advanced chapter of category theory which aren't discussed in the previous books.

F. Borceux: Handbook of Categorical Algebra 1: Basic Category Theory

F. Borceux: Handbook of Categorical Algebra 2: Categories and Structures

F. Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves

For higher category theory I know just few reference:

Leinster's "Higher Operads Higher Categories" (http://arxiv.org/abs/math/0305049),

and

Lurie's "Higher Topos Theory" (http://arxiv.org/abs/math/0608040)

other good reference in higher category theory and higher dimensional algebra in general are Baez'This week's finds and arxiv articles Higher dimensional algebra*.

math.CT
QA
advice
S. MacLane: Category theory for working mathematicians (I've already said a lot about this)

S. Awodey: Category theory (Peculiar because it has very low prerequisites and it's rich of examples too)

J. Adamek,H. Herrlich, G. Strecker: Abstract and concrete category theory (freely avaible at at this site "http://katmat.math.uni-bremen.de/acc/acc.pdf", maybe the book with the greatest number of examples from topology and algebra)

After you have read one of these book, you could also use Borceux's books and read some more advanced chapter of category theory which aren't discussed in the previous books.

F. Borceux: Handbook of Categorical Algebra 1: Basic Category Theory

F. Borceux: Handbook of Categorical Algebra 2: Categories and Structures

F. Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves

For higher category theory I know just few reference:

Leinster's "Higher Operads Higher Categories" (http://arxiv.org/abs/math/0305049),

and

Lurie's "Higher Topos Theory" (http://arxiv.org/abs/math/0608040)

other good reference in higher category theory and higher dimensional algebra in general are Baez'This week's finds and arxiv articles Higher dimensional algebra*.

august 2018 by quamash

Readings and Assignments | Category Theory for Scientists | Mathematics | MIT OpenCourseWare

august 2018 by quamash

Baez, John C., and Mike Stay. "Physics, Topology, Logic, and Computation: A Rosetta Stone." March 2, 2009.

Gromov, Misha. "This resource may not render correctly in a screen reader.In a Search for a Structure, Part 1: On Entropy." (PDF) July 10, 2012.

Weinstein, Alan. "Groupoids Unifying Internal and External Symmetry: A Tour through Some Examples." Notices of the American Mathematical Society 43 (1996): 744–52.

math.CT
Gromov, Misha. "This resource may not render correctly in a screen reader.In a Search for a Structure, Part 1: On Entropy." (PDF) July 10, 2012.

Weinstein, Alan. "Groupoids Unifying Internal and External Symmetry: A Tour through Some Examples." Notices of the American Mathematical Society 43 (1996): 744–52.

august 2018 by quamash

A Neighborhood of Infinity: A Partial Ordering of some Category Theory applied to Haskell

august 2018 by quamash

I've had a few requests from people wanting to teach themselves applications of Category Theory to Haskell based on posts I've made. I've made things difficult by posting stuff at random levels of difficulty and without any kind of organising thread through them.

math.CT
august 2018 by quamash

Haskell/Category theory - Wikibooks, open books for an open world

august 2018 by quamash

A category is, in essence, a simple collection. It has three components:

A collection of objects.

A collection of morphisms, each of which ties two objects (a source object and a target object) together. (These are sometimes called arrows, but we avoid that term here as it has other connotations in Haskell.) If f is a morphism with source object A and target object B, we write f : A → B {\displaystyle f:A\to B} f:A\to B.

A notion of composition of these morphisms. If g : A → B {\displaystyle g:A\to B} g:A\to B and f : B → C {\displaystyle f:B\to C} f:B\to C are two morphisms, they can be composed, resulting in a morphism f ∘ g : A → C {\displaystyle f\circ g:A\to C} f\circ g:A\to C.

math.CT
A collection of objects.

A collection of morphisms, each of which ties two objects (a source object and a target object) together. (These are sometimes called arrows, but we avoid that term here as it has other connotations in Haskell.) If f is a morphism with source object A and target object B, we write f : A → B {\displaystyle f:A\to B} f:A\to B.

A notion of composition of these morphisms. If g : A → B {\displaystyle g:A\to B} g:A\to B and f : B → C {\displaystyle f:B\to C} f:B\to C are two morphisms, they can be composed, resulting in a morphism f ∘ g : A → C {\displaystyle f\circ g:A\to C} f\circ g:A\to C.

august 2018 by quamash

Table of contents—The Stacks project

august 2018 by quamash

Part 1: Preliminaries

Chapter 1: Introduction

Chapter 2: Conventions

Chapter 3: Set Theory

Chapter 4: Categories

Chapter 5: Topology

Chapter 6: Sheaves on Spaces

Chapter 7: Sites and Sheaves

Chapter 8: Stacks

Chapter 9: Fields

Chapter 10: Commutative Algebra

Chapter 11: Brauer groups

Chapter 12: Homological Algebra

Chapter 13: Derived Categories

Chapter 14: Simplicial Methods

Chapter 15: More on Algebra

Chapter 16: Smoothing Ring Maps

Chapter 17: Sheaves of Modules

Chapter 18: Modules on Sites

Chapter 19: Injectives

Chapter 20: Cohomology of Sheaves

Chapter 21: Cohomology on Sites

Chapter 22: Differential Graded Algebra

Chapter 23: Divided Power Algebra

Chapter 24: Hypercoverings

math.AG
math.AT
math.CT
Chapter 1: Introduction

Chapter 2: Conventions

Chapter 3: Set Theory

Chapter 4: Categories

Chapter 5: Topology

Chapter 6: Sheaves on Spaces

Chapter 7: Sites and Sheaves

Chapter 8: Stacks

Chapter 9: Fields

Chapter 10: Commutative Algebra

Chapter 11: Brauer groups

Chapter 12: Homological Algebra

Chapter 13: Derived Categories

Chapter 14: Simplicial Methods

Chapter 15: More on Algebra

Chapter 16: Smoothing Ring Maps

Chapter 17: Sheaves of Modules

Chapter 18: Modules on Sites

Chapter 19: Injectives

Chapter 20: Cohomology of Sheaves

Chapter 21: Cohomology on Sites

Chapter 22: Differential Graded Algebra

Chapter 23: Divided Power Algebra

Chapter 24: Hypercoverings

august 2018 by quamash

Confusion about Homotopy Type Theory terminology - Mathematics Stack Exchange

june 2018 by quamash

The usual names for Σ-types and Π-types are dependent sum and dependent product, respectively, but for some reason the Homotopy Type Theory book calls them dependent pair type and dependent function type.

QA
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cs.Pl
june 2018 by quamash

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