Questions tagged [ct.category-theory]

Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

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Linear algebra in terms of abstract nonsense?

The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think. I was wondering what portions of basic linear algebra (first couple of courses) fall ...
Arrow's user avatar
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What parts of the theory of quasicategories have been simplified since the publication of HTT?

It has been almost ten years since Lurie published Higher Topos Theory, where (following Joyal and probably others) he set up foundations for higher category theory via quasicategories. My impression ...
dhy's user avatar
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Invertible matrices of natural numbers are permutations... why?

I have heard the following statement several times and I suspect that there is an easy and elegant proof of this fact which I am just not seeing. Question: Why is it true that an invertible nxn ...
Chris Schommer-Pries's user avatar
38 votes
7 answers
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What is DAG and what has it to do with the ideas of Voevodsky?

In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ...
veit79's user avatar
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What's a reasonable category that is not locally small?

Recall that a category C is small if the class of its morphisms is a set; otherwise, it is large. One of many examples of a large category is Set, for Russell's paradox reasons. A category C is ...
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A “mother of all groups”? What kind of structures have "mother of all"s?

For ordered fields, we have a “mother of all ordered fields”, the surreal numbers $\mathbf{No}$, a proper-class “field” which includes (an isomorphic copy of) every other ordered field as a subfield. ...
The_Sympathizer's user avatar
38 votes
8 answers
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Geometric intuition for limits

I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects ...
Charles Staats's user avatar
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6 answers
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Advantages of diffeological spaces over general sheaves

I have been playing with/thinking about diffeological spaces a bit recently, and I would like understand something rather crucial before going further. First a little background: Diffeological spaces ...
David Carchedi's user avatar
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3 answers
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What does the term "yoga" mean in mathematics?

Just exactly what the title says; often, in mathematics, particularly in the vicinity of Grothendieck, I see reference to "the yoga of...". What exactly does the term "yoga" mean in these contexts?
Sridhar Ramesh's user avatar
38 votes
4 answers
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Completion of a category

For a poset $P$ there exists an embedding $y$ into a complete and cocomplet poset $\hat{P}$ of downward closed subsets of $P$. It is easy to verify that the embedding preserves all existing limits and ...
Michal R. Przybylek's user avatar
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How do you define (infinity,1) categories in Homotopy Type Theory?

One of the major motivations of Homotopy Type Theory is that it naturally builds in higher coherences from the beginning. One important setting where higher coherence requirements get annoying is ...
Noah Snyder's user avatar
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In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?

In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra. He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie ...
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3 answers
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Why are profinite topologies important?

I hope this is not too vague of a question. Stone duality implies that the category Pro(FinSet) is equivalent to the category of Stone spaces (compact, Hausdorff, totally disconnected, topological ...
Mike Shulman's user avatar
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4 answers
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How many morphisms from 1 to 1+1 can there be?

Here is an interesting question raised by Alice Rhyl. Let $C$ be a category with a terminal object $1$ and finite coproducts. How many different morphisms $f : 1 \to 1 + 1$ can there be? There are ...
John Baez's user avatar
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How can I define the product of two ideals categorically?

Given a commutative ring $R$, there is a category whose objects are epimorphisms surjective ring homomorphisms $R \to S$ and whose morphisms are commutative triangles making two such epimorphisms ...
Qiaochu Yuan's user avatar
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7 answers
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Limits in category theory and analysis

Is it possible to regard limits in analysis (say, of real sequences or more generally nets in topological spaces) as limits in category theory? Is there some formal connection? Edit ('13): Perhaps it ...
Martin Brandenburg's user avatar
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13 answers
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Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful? I ...
Lehs's user avatar
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Could groups be used instead of sets as a foundation of mathematics?

Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The ...
Oscar Cunningham's user avatar
37 votes
4 answers
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When is the opposite of the category of algebras of a Lawvere theory extensive?

When is the opposite of the category of algebras of a Lawvere theory an extensive category? Any necessary or sufficient conditions on the Lawvere theory will be interesting to me. Here's why I'm ...
John Baez's user avatar
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1 answer
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Are there any nontrivial abelian categories with only finitely many objects?

The title says pretty much what I want. Of course, the abelian categories should contain at least one nonzero object. In particular, is there an abelian category containing only one nonzero object? ...
user27976's user avatar
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5 answers
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Locales and Topology.

As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...
37 votes
3 answers
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Does the category of (algebraically closed) fields of characteristic $p$ change when $p$ changes?

Let $\mathrm{ACF}_p$ denote the category of algebraically closed fields of characteristic $p$, with all homomorphisms as morphisms. The question is: when is there an equivalence of categories between $...
Tim Campion's user avatar
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4 answers
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What other monoidal structures exist on the category of sets?

I know of the following monoidal structures over $\mathbf{Set}$ (taken from here): The Cartesian product $\otimes=\times$ (categorical product) The disjoint union $\otimes=+$ (categorial coproduct) $...
JMAA's user avatar
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3 answers
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What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...
Keshav Srinivasan's user avatar
37 votes
3 answers
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What is the point of pointwise Kan extensions?

Recall that a Kan extension is called pointwise if it can be computed by the usual (co)limit formula, or equivalently if it is preserved by (co)representable functors. I have seen pointwise Kan ...
Tim Campion's user avatar
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What about stacks of categories in algebraic geometry?

Stacks qua moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The ...
David Roberts's user avatar
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What is the meaning of this analogy between lattices and topological spaces?

Let me add one more edit to help explain why this is a serious question. Theorem 5 below is a sort of lattice version of Urysohn's lemma, and it has essentially the same proof. Theorem 6, the famous ...
Nik Weaver's user avatar
37 votes
0 answers
1k views

Computer calculations in A_infinity categories?

Is there a good computer program for doing calculations in A-infinity categories? Explicit calculations in A-infinity categories are an important, useful, yet very tedious task. One has to keep track ...
Heinrich Hartmann's user avatar
36 votes
11 answers
10k views

Categories First Or Categories Last In Basic Algebra?

Recently, I was reminded in Melvyn Nathason's first year graduate algebra course of a debate I've been having both within myself and externally for some time. For better or worse, the course most ...
36 votes
6 answers
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The category of posets

I am trying to teach myself category theory and, as a beginner, I am looking for examples that I have a hands-on experience with. Almost every introductory text in category theory contains following ...
36 votes
3 answers
6k views

Conjectures in Grothendieck's "Pursuing stacks"

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to ...
AAK's user avatar
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36 votes
3 answers
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"Softness" vs "rigidity" in Geometry

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually,...
Qfwfq's user avatar
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36 votes
3 answers
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Tannaka formalism and the étale fundamental group

For quite a while, I have been wondering if there is a general principle/theory that has both Tannaka fundamental groups and étale fundamental groups as a special case. To elaborate: The theory of ...
Lars's user avatar
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36 votes
1 answer
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Is there a general theory of "compactification"?

In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
Tim Campion's user avatar
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36 votes
2 answers
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What are all the natural maps between iterated duals of vector spaces, and equations between these?

Fix a field. Given a vector space $V$ it has a dual $V^\ast$, which has its own dual $V^{\ast \ast}$, which has its own dual $V^{\ast \ast \ast}$, and so on ad infinitum. There are lots of natural ...
John Baez's user avatar
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36 votes
1 answer
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Infinite tensor products

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
Martin Brandenburg's user avatar
35 votes
12 answers
3k views

No canonical isomorphism [duplicate]

I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...
35 votes
9 answers
14k views

What is... a grossone?

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The ...
Mikhail Katz's user avatar
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35 votes
5 answers
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Understanding a quip from Gian-Carlo Rota

In the chapter "A Mathematician's Gossip" of his renowned Indiscrete Thoughts, Rota launches into a diatribe concerning the "replete injustice" of misplaced credit and "forgetful hero-worshiping" of ...
35 votes
3 answers
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Higher Topos Theory- what's the moral?

I've often seen Lurie's Higher Topos Theory praised as the next "great" mathematical book. As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems ...
Michael Klyachman's user avatar
35 votes
2 answers
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Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"?

This is a follow-up to Dan Ramras' answer of this question. The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here). The weak Hausdorff rather ...
archipelago's user avatar
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35 votes
3 answers
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Does the hypergraph structure of the set of subgroups of a finite group characterize isomorphism type?

Question Suppose there is a bijection between the underlying sets of two finite groups $G, H$, such that every subgroup of $G$ corresponds to a subgroup of $H$, and that every subgroup of $H$ ...
Chris Beck's user avatar
35 votes
3 answers
2k views

Internal logic of the topos of simplicial sets

I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, ...
Mike Shulman's user avatar
35 votes
3 answers
6k views

What is a triangle?

So I've been reading about derived categories recently (mostly via Hartshorne's Residues and Duality and some online notes), and while talking with some other people, I've realized that I'm finding it ...
Will Chen's user avatar
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35 votes
2 answers
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Is the fundamental group functor a left-adjoint?

Theorem 1B.9 in Hatcher's Algebraic Topology says that for a (pointed) connected CW complex $X$ and group $G$, there is a bijection $\text{Hom}(\pi_1(X), G) \cong [X,K(G,1)]$, where $\pi_1(X)$ is the ...
ziggurism's user avatar
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35 votes
3 answers
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Is every abelian group a colimit of copies of Z?

More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$? Note that this does not follow ...
Qiaochu Yuan's user avatar
35 votes
4 answers
4k views

An intelligent ant living on a torus or sphere – Does it have a universal way to find out?

I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a ...
Claus's user avatar
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34 votes
8 answers
3k views

Big list of comonads

The concept of a monad is very well established, and there are very many examples of monads pertaining almost all areas of mathematics. The dual concept, a comonad, is less popular. What are ...
34 votes
1 answer
5k views

Freyd-Mitchell's embedding theorem

Freyd–Mitchell's embedding theorem states that: if $A$ is a small abelian category, then there exists a ring R and a full, faithful and exact functor $F\colon A \to R\text{-}\mathrm{Mod}$. I have been ...
Bruno Stonek's user avatar
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34 votes
4 answers
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Surveys of Goodwillie Calculus

Is there a good general introduction to Goodwillie calculus out there, like a paper or publication that gives a general overview of the calculus as well as how it is useful and why we are interested ...
Jonathan Beardsley's user avatar