# Questions tagged [ct.category-theory]

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

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### On the universal property for interval objects

In his lecture, The Categorical Origins of Lebesgue Measure, Professor Tom Leinster mentions the following theorem:
Theorem 1: (Freyd; Leinster) The topological space $[0, 1]$ comes equipped with ...

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### Need sheaves take value in small categories?

In all the definitions of (pre)sheaves that I have seen, they are always contravariant functors from some category $\mathcal{C}$ to a small category, say, Set. However, the definitions are always ...

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### What are the (co)algebras for the $(\operatorname{Hom}(A,-), A\otimes-)$ adjunction (co)monad?

A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any ...

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### Elementary example of right localization of functor

I am learning about a general framework for derived functors from Hotta et al., D-modules, Perverse Sheaves, and Representation Theory, Appendix B.
$\newcommand{\CC}{\mathcal C} \newcommand{\DD}{\...

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### Definitions of hypercovers by generalized cover

Let $f: E\to B$ be a map between presheaves of sets. One says that $f$ is a generalized cover if given any map $rX\to B$, there is a covering sieve $R\hookrightarrow rX$ such that for every element $...

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### By general reasons, $i_A \colon \mathbb{D}\text{-}\mathrm{cont}[A,\mathbf{Set}] \to [A,\mathbf{Set}]$ has a left adjoint

In Centazzo and Vitale's A Duality Relative to a Limit Doctrine (TAC, 2002, abstract), early on, they make the above claim and cite Kelly's Basic Concepts in Enriched Category Theory (TAC reprints). I ...

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159 views

### CG spaces from the perspective of sheaves over compact Hausdorff spaces

A compactly generated space is a space $X$ such that $f : X \rightarrow Y$ is continuous if and only if $K \rightarrow X \stackrel{f}{\rightarrow} Y$ is continuous for each compact hausdorff space $K$....

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202 views

### Motivation/intuition behind the definition of delta-functors and related concepts

I originally posted this on Maths SE, but then realised that the question probably fits MO better, as my objective was to gain different perspectives regarding the matter.
Why are $\delta$-functors ...

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844 views

### Who invented Monoid?

I was trying to find (and failed) the original author of either
the concept of Monoid (set with binary associative operation and identity)
the name (which sounds french ? and also Dioid (for what ...

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93 views

### Structure of a poset of subcategories

Given a category $\mathbf{C}$, we can consider monomorphisms into it. These are the faithful and injective-on-objects functors (this violates the principle of equivalence). The idea is to try to get a ...

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### Eventually non vanishing tors

Let $A$ be a commutative $k$-algebra, for $k$ a field of characteristic $0$. Let $Perf_{A}$ denote the dg category of cohomologically graded $A$-modules and let $M\in Perf_{A}$ be a classical perfect ...

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113 views

### Understanding the reason for the particular formulation of the definition of a concrete reflector (as stated in The Joy of Cats)

This question is essentially a followup of this question. But before going into the question let me introduce the relevant definitions as given in The Joy of Cats.
Definition 1. Let $\bf{X}$ be a ...

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252 views

### When do triples have been called monads for the first time?

I am fine-tuning a short note on basic category theory; any such course must introduce monads, and I want to give a bit of history of the subject.
I soon realized that I don't know the precise series ...

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135 views

### Proving a Kan-like condition for functors to model categories?

I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...

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187 views

### Why does every chain complex have a map into its cone?

In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...