# Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seeming different areas of mathematics.

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### Categorical introduction to Algebra and Topology

At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book Algebra by Serge Lang. I have read the chapters on groups and rings, but then my ...
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### What's the deal with empty models in first-order logic?

Asaf's answer here reminded me of something that should have been bothering me ever since I learned about it, but which I had more or less forgotten about. In first-order logic, there is a convention ...
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### Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

Under which name is the following proposition filed actually: Every poset $P$ embeds fully and faithfully in the powerset of $P$, ordered by subset inclusion. Let me call it Dedekind's lemma. ...
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### Motivation and use for category theory?

From reading the answers to different questions on category theory, it seems that category theory is useful as a framework for thinking about mathematics. Also, from the book Algebra by Saunders Mac ...
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### Honest application of category theory

I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a ...
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### what is the difference between functor and function?

As it is, what is the difference between functor and function? As far as I know, they look really similar. And is functor used in set theory? I know that function is used in set theory. Thanks.
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### Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
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### What does “homomorphism” require that “morphism” doesn't?

I'm starting to learn category theory, but there's one thing I don't get: all morphisms seem to be homomorphisms; the definition seems to be the same. What's the difference between these two? Can you ...
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### Concrete examples of 2-categories

I've been reading some of John Baez's work on 2-categories (eg here) and have been trying to visualize some of the constructions he gives. I'm interested in coming up with 'concrete' examples of 2-...
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### Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
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### 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 ...
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### Analogue of the Cantor-Bernstein-Schroeder theorem for general algebraic structures

The Cantor-Bernstein-Schroeder theorem states that if there are two sets $A$ and $B$ such that there exist injective (alternatively, surjective, assuming choice I think) maps $A \to B$ and $B \to A$, ...
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### For which categories we can solve $\text{Aut}(X) \cong G$ for every group $G$?

It is usually said that groups can (or should) be thought of as "symmetries of things". The reason is that the "things" which we study in mathematics usually form a category and for every object $X$ ...
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### Why is every category not isomorphic to its opposite?

This is a beginner category theory question: I'm trying to wrap my head around the fact that we do not have $\mathbf{Sets} \cong \mathbf{Sets}^\mathsf{op}$, i.e., the category of sets is not ...
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### Examples of categories where morphisms are not functions

Can someone give examples of categories where objects are some sort of structure based on sets and morphisms are not functions?
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### Uniqueness of adjoint functors up to isomorphism

Suppose we are given functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G,G':\mathcal{D}\rightarrow\mathcal{C}$ such that $G$ and $G'$ are both right adjoint to $F$. To show that $G$ and $G'$ are ...
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### Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over $\mathbb{R}$...
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### Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
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### Epimorphism and Monomorphism = Isomorphism?

It seems to be that if a map is both an epimorphism and a monomorphism, it is not necessarily the case that it is an isomorphism. However, in the category of sets, if a map is both an epimorphism and ...