Search Results

I don't believe there are non-trivial examples of this concept. Assume that $e: X \to X$ is a constant map, then $e$ is idempotent. Any category can be embedded fully faithfully into a Cauchy complete …

Firstly, note that it is enough to construct an isomorphism $$\mathcal{C}(L(A,B),C) \simeq \mathcal{B}(B, R_2(A,C))$$ The third natural isomorphism then follows automatically. Secondly, standard appli …

We have an equivalence of categories $Aff\simeq Ring^{op}$ and a pair of adjoint functors $$\mathcal{O}:Sch\rightleftharpoons Ring^{op} : Spec$$ $$\mathcal{O} \dashv Spec$$ The category of affine sche …

The definition of object being initial is obviously equivalent to the statement "the inclusion of one-point category is final". Here by final functor I mean $p: I\to C$, such that for any $F: C\to D$ …

The method of forcing in mathematical logic. If you want to prove the consistency of axiom systems, you can just explicitly present a model of it. To prove results about set theory itself, like the i …

I prefer a somewhat different view of ends and coends, with the intuition stemming more from classical linear algebra and functional analysis. So for me an end is really an integral in a categorical s …

Consider a theory which has no models in $\mathrm{Set}$, but has a model in $\mathrm{Sh}(L)$ for some locale $L$. For example, the theory $\mathcal{CLF}$ of complete linearly ordered fields with more …

There's an enlightening example of limit coming from topology. Arguably it was one of the motivating examples for the notion of categorical limit. In general topology it is known as limit over a filte …

I like to view Yoneda's lemma as a generalization of the description of Galois coverings in topology. To any functor $F: C \to Set$ we can associate its category of elements $El(F)$. Its objects are p …

There is a purely category-theoretical way to describe affine schemes. Consider an arbitrary scheme $X$. For any ring $R$ we can consider a set of $R$-points of $X$, that is $X(R)=\mathrm{Hom}_{Schm}\ …

It is a general fact that if you consider an abelian group $A$ in topos $T$, then equivalence classes of $A$-torsors in $T$ are classified by cohomology group $H^1(T;A)=Ext^1_T(\mathbb{Z},A)$. Here $\ …

Tom Leinster's book is very old. For higher category theory 2003 is like a previous epoch. In those times there were many competing definitions of higher category theory and higher algebra, for most o …

I was talking about the following tentative argument. The 2-category of distributors (also called profunctors) $\mathrm{Dist}$ has (small) categories for objects. For $C,D:\mathrm{Dist}$ the 2-categor …

In a nutshell, the question is: is it true that any explicit (not involving axiom of choice) pointwise transformation between sufficiently complicated functors is natural almost everywhere? Let $C$ …

As requested, making comment into the answer. I think Helemskii's book "Lectures and Exercises on Functional Analysis" contains a very nice intro into category theory. It may be a bit light on the alg …