1
vote
1answer
87 views

Foundations book using category theory?

I'm about to embark on a PhD in mathematical biology. My major is in computer science. I would like to acquire a more rigorous understanding of math, which I am going to need to tackle some research ...
0
votes
0answers
32 views

Module and bimodule categories equivalent to a 2-functor

Let $A$ be a tensor category (i.e. monoidal category). A (right) module category of $A$ is a category $M$ with a coherent action $\mu\colon M\otimes A\to M$. Denote $BA$ be the one-object bicategory ...
1
vote
0answers
40 views

coalgebra/algebra of the identity endfunctor

Let $\mathbb{C}$ be a small/locally small category and let $T:\mathbb{C} \to \mathbb{C}$ be an endofunctor. One can then have $T$-algebras and $T$-coalgebras in the usual way: for $X,Y \in ...
3
votes
3answers
79 views

From groups to groupoids.

Let $\mathcal{G}$ be a groupoid and $p$ an object in $\mathcal{G}.$ It is well known that the set ${\rm Mor}_{\mathcal{G}}(p,p)$ is a group. I would like to know if there is a way to recognize a ...
4
votes
1answer
100 views

iterated dual vector spaces

Let $K$ be a field and $\mathcal U$ a universe such that $K\in\mathcal U$. (Here, "universe" means "uncountable Grothendieck universe".) Let $\mathcal C$ be the category of $K$-vector spaces belonging ...
3
votes
4answers
494 views

Objects are finite sets, arrows are matrices. How is this a category?

I just started to read this book on category theory. How is this example below a category? I have difficulty imagining what this construct really is. Could someone please illuminate me ? I ...
3
votes
0answers
37 views

Logic in closed symmetric monoidal categories; reference request.

Suppose we want an algebraic theory $T$ to be interpretable in any closed symmetric monoidal category $\mathbf{C}.$ I am thinking in particular of the case where $\mathbf{C}$ is the category of models ...
6
votes
3answers
170 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
3
votes
3answers
119 views

Some reference for categorical logic?

By "categorical logic" I mean category-theoretical models of logic. In particular, I am more interested in models of intuitionistic predicate logic with conjunction, disjunction, implication and ...
0
votes
0answers
25 views

An interleaved path of arrows in a category/digraph

In what I am doing now, a slightly strange concept emerged; My objects are sequences of arrows $\{f_i\}_{i=1}^{n}$ in a small category such that for every $1\leq i\leq n-1$, there is an arrow from the ...
4
votes
0answers
90 views

adjoints and duality

Let $X$ and $Y$ be posets seen as a category. Let $F:X \to Y$ and $G: Y \to X$ be an adjunction. That is, $(G \circ F) (x) \ge_X x$ and $(F \circ G) (y) \le_Y y$. Now, $F$ is additive and $G$ is ...
0
votes
1answer
20 views

A class of “internal” endofunctors in cartesian closed categories

I'm interested in a class of endofunctors on cartesian closed categories with a quite natural definition, and am wondering whether/where this class has been studied so far (and how it is called). Fix ...
2
votes
0answers
12 views

Internal additions in additive categories agree with given ones

I know that if $\mathcal C$ is a semiadditive category, then for every two objects $A$, $B$, the set $\mathrm{Hom}_\mathcal{C} (A, B)$ is automatically endowed with a structure of commutative monoid, ...
0
votes
0answers
137 views

Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
2
votes
2answers
35 views

Distributivity in linear monoidal categories

Let $\mathcal{C}$ be a linear monoidal category, that is a monoidal category (with tensor product $\otimes$) enriched over $\mathbf{Vect}$. Now as far as I can tell the axioms for a linear category ...
3
votes
0answers
45 views

Inductive limit of manifolds?

The inductive limit of a direct system of manifolds is a topological space (which I don't think needs be a manifold). But it seems like it should retain some of the structure of manifolds : for ...
2
votes
0answers
48 views

The composition of functors and algebraic structures

An algebraic structure can be viewed as a finite-product-preserving functor $X : \mathbf{L} \rightarrow \mathbf{C},$ where $\mathbf{L}$ is a Lawvere theory and $\mathbf{C}$ is a category. Thus given ...
2
votes
1answer
33 views

“Topologification” of a subcollection of a power set

Let $X$ be any set and consider any $\mathscr{S} \subseteq \mathcal{P}(X)$, where the latter is the power set. It is natural to ask if we make $X$ a topological space by the subcollection ...
4
votes
1answer
53 views

Where does one learn the algebraic geometry needed for topos theory?

I am a masters student familiar with category theory. I have started learning topos theory from MacLane-Moerdijk's book "Sheaves in Geometry and Logic: A First Introduction to topos Theory". I get the ...
4
votes
0answers
61 views

Where can I learn more about the concept that is dual to “relation”?

Let $X$ and $Y$ denote sets. Then a relation $X \rightarrow Y$ is, by definition, a subset of $X \times Y$. Dually, we can define that a "corelation" $X \rightarrow Y$ is a partitioning of $X \uplus ...
6
votes
0answers
112 views

Cantor-Schröder-Bernstein without elements

The Cantor-Schröder-Bernstein Theorem for the category of sets has several "analogues" for other categories as well, for example measurable spaces and operator algebras (but not for arbitrary ...
5
votes
1answer
68 views

The infinite Direct Sum in the category Ring

If you don't have strong personal feelings about it already, most of you have at least witnessed the opposing factions on how we should define a ring and, by extension, how we should define a ring ...
2
votes
1answer
50 views

Prove a categorical statement

In the answer to Direct products in subcategories it is said: If $\mathcal{D}$ is a full subcategory of $\mathcal{C}$ and $A \times_{\mathcal{C}} B$ is (isomorphic to) an object of $\mathcal{D}$, ...
1
vote
0answers
41 views

How much does a theory tell about categorical properties of its models.

Let take a first order theory $T$ in a language $\mathcal L$. One can form the category $\mathrm{Mod}(T)$ whose objects are the models of $T$ (those $\mathcal L$-structures $\mathfrak M$ such that ...
5
votes
0answers
55 views

Best approximation to an adjoint functor

I have the following question. Suppose I have a functor $F\colon C\to D$ between two categories. I would like it to have an adjoint (say, right), but it doesn't. Is there a way to define a "best ...
7
votes
3answers
304 views

Theorems implied by Yoneda's lemma?

Ok, so I was reading the Wikipedia article on Yonedas lemma. And I've heard before that when you prove things in category theory you automatically get a lot of results by proving it in abstract ...
2
votes
0answers
66 views

Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
16
votes
4answers
358 views

A Category Theoretical view of the Kalman Filter

Some basic background The Kalman filter is a (linear) state estimation algorithm that presumes that there is some sort of uncertainty (optimally Gaussian) in the state observations of the dynamical ...
4
votes
2answers
140 views

Set theory aspects of category theory

I have never learnt axiomatic set theory, but have studied it from Munkres's Topology book first chapter. So I do not understand the difference between a class and a set except in some vague sense. ...
2
votes
1answer
51 views

Algebraic signatures as quivers; is there somewhere I can learn more about these definitions?

In my opinion, a cool definition of "algebraic signature" is as follows: An algebraic signature on the sort symbols $\mathcal{X} = \{X_0,...,X_{n-1}\}$ is precisely a quiver whose underlying set ...
3
votes
0answers
44 views

Seeking information about (category-theoretic) varieties, quasivarieties, and universal Horn classes.

I'm looking for a list of basic facts regarding (category-theoretic) varieties, quasivarieties, and universal Horn categories, as well as information about which forgetful functors preserve what. In ...
4
votes
1answer
76 views

Has anyone successfully axiomatized the category of finite sets? In such a way that the resulting theory is bi-interpretable with PA.

In studies of ZFC, it is conventional to take Peano arithmetic (hereafter PA) as the metatheory. However, I don't like this convention; I think a better approach would be a metatheory (like ZF-fin ...
6
votes
1answer
74 views

Higher categorical Yoneda lemma

One of the most powerful results in ordinary category theory is the Yoneda lemma, and so the following question seems natural: Is there an analogue of the Yoneda lemma for (weak) $n$-categories? I ...
3
votes
1answer
51 views

The category of models of a commutative algebraic theory.

I'd like more information about the category of models $\mathcal{C}$ of a commutative algebraic theory. In particular: Do finite coproducts necessarily exist, and if so, do they necessarily coincide ...
8
votes
1answer
188 views

Advice: Algebra and category theory for geometry?

I'm interested in learning a bit of geometry. To start I'm (slowly) working my way towards differential geometry via Lee's Introduction to Smooth Manifolds. But, later on, I'd also like to study some ...
9
votes
2answers
193 views

Introduction to sheaves using categorical approach

When I first started to learn about sheaves, it was a very geometric approach. This is nice, but it seems like knowing more abstract categorical approach is very useful. For example, sheafification ...
17
votes
1answer
356 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
3
votes
1answer
41 views

Where can I learn more about these subcategories of functor categories?

Note: I've substantially edited the definition; $f$ is now allowed to be a functor. Given categories $\mathcal{C}$ and $\mathcal{D}$, we can form the functor category $[\mathcal{C},\mathcal{D}]$. Now ...
2
votes
2answers
54 views

When the homsets of a category are structured.

If the objects of a category are algebraic structures in their own right, this often places additional structure on the homsets. Is there somewhere I can learn more about this general idea? Example ...
5
votes
1answer
95 views

Advice regarding best-practice mathematics / categorial logic.

A good heuristic is: If it doesn't cost anything, generalize. In particular, if we have a theorem, and a proof thereof, then we ought to look for a maximal generalization of this theorem, ...
0
votes
2answers
89 views

The Adjunction $\_\times A\dashv (\_ )^A$ for Preorders: The Deduction Theorem.

The following is from Turi's Category Theory Lecture Notes. Definition 11.11 Let $A$ be an object of a category $\mathbb{C}$ with binary products. The right adjoint of $\_\times ...
10
votes
1answer
170 views

Applications of TQFTs beyond physics

I'm giving a talk at a postgrad seminar on the topic of topological quantum field theories (TQFTs) with a mixed audience of pure and applied mathematicians. As such, I'd like to be able to offer some ...
5
votes
1answer
57 views

Exact sequence in a category with zero morphisms

Let $C$ be a category with zero morphisms (equivalently, $\mathsf{Set}_*$-enriched), for example it could be a linear category. Then we can talk about kernels and cokernels of morphisms in $C$. I ...
7
votes
5answers
145 views

Deeper studies in Category Theory: suggestions and references.

I discovered Category Theory one year and a half ago and I got addicted. I studied some of the basic concepts of this branch (limits, adjunctions etc.), including some sightseeing into abelian ...
2
votes
1answer
37 views

Reference request for 2-category theory

In the theory of 2-categories, we have the following theorem. A pseudofunctor is an equivalence of 2-categories if it is essentially surjective on objects, and the induced functor on Hom categories ...
4
votes
1answer
120 views

Dual and adjoint operator

Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual ...
7
votes
2answers
190 views

How much Category theory one must learn?

I have learnt very basic category theory (up to Yoneda lemma from Hungerford's Algebra text). My question is how much category theory should every Mathematics student who is not planning to specialize ...
5
votes
1answer
142 views

Volume 3 of Johnstone's “Sketches of an Elephant”

Recently, I read the Chapter 8 of Johnstone's "Topos theory" and got interested in the homotopy and cohomology theory of Grothendieck toposes. So I'm looking for the textbooks expanding these ...
3
votes
0answers
57 views

Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
2
votes
1answer
42 views

Familiarizing with the Grothendieck topos $\mathbf{B}G$.

I am trying to familiarize with the Grothendieck topos $\mathbf{B}G$ of continuous $G$-sets, where $G$ is a topological group. I am unfortunately not very familiar with working with different ...