ﻻ يوجد ملخص باللغة العربية
For topological spaces $X$ and $Y$, a (not necessarily continuous) function $f:X rightarrow Y$ naturally induces a functor from the category of closed subsets of $X$ (with morphisms given by inclusions) to the category of closed subsets of $Y$. The function $f$ also naturally induces a functor from the category of closed subsets of $Y$ to the category of closed subsets of $X$. Our aim in this expository note is to show that the function $f$ is continuous if and only if the first of the above two functors is a left adjoint to the second. We thereby obtain elementary examples of adjoint pairs (apparently) not part of the standard introductory treatments of this subject.
Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $infty$-categories. One of our main results is an $infty$-categorical ge
In the context of infinity categories, we rethink the notion of derived functor in terms of correspondences. This is especially convenient for the description of a passage from an adjoint pair (F,G) of functors to a derived adjoint pair (LF,RG). In p
We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $infty$-categories -- in particular, these $n$-categories do not admit all small (co)l
We show that the comma category $(mathcal{F}downarrowmathbf{Grp})$ of groups under the free group functor $mathcal{F}: mathbf{Set} to mathbf{Grp}$ contains the category $mathbf{Gph}$ of simple graphs as a full coreflective subcategory. More broadly,
Let $mathcal{B}$ be a subcategory of a given category $mathcal{D}$. Let $mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $mathcal{B}$ to $mathcal{D}$ such that $mathcal{B}$ becomes a sub m