We show that the cartesian closed category of compactly generated Hausdorff spaces is regular, but is neither exact, nor locally cartesian closed. In fact we find a coequalizer of an equivalence relation which is not stable under pullbacks.

A proper short exact sequence 0! A!f B! C ! 0 ð*Þ in the category of locally compact abelian groups is said to be t-pure if fðAÞ is a topologically pure subgroup of B, that is, if

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE. More generally we show that K1(W) of a Waldhausen category W with cylinders and a saturated class of weak equivalences agrees with K1(DW) of the associated right pointed derivator DW. Introduction For a long time there was an interest in defining a nice K-theory for triangulated categories ...

In this paper, we first define the notion of $\mathcal{F}$-cosmall quotient for an additive exact substructure $\mathcal{F}$ structure $\mathcal{E}$ in category $\mathcal{A}$. We show that every is right minimal some cases. also give definition $\mathcal{F}$-superfluous and relate it approximation modules. As application, investigate our results a pure-exact $\mathcal{F}$.

In analogy with the relation between closure operators in presheaf toposes and Grothendieck topologies, we identify the structure in a category with finite limits that corresponds to universal closure operators in its regular and exact completions. The study of separated objects in exact completions will then allow us to give conceptual proofs of local cartesian closure of different categories ...

Relative theories(=closed subfunctors) are considered in exact, triangulated and extriangulated categories by Dräxler-Reiten-Smalø-Solberg-Keller, Beligiannis Herschend-Liu-Nakaoka, respectively. We give a construction method of closed subfunctors from given half exact functors which contains existing constructions. Moreover, if an category has enough projective objects, then every subfunctor i...

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming t...

We develop in this paper a stable theory for projective complexes, by which we mean to consider chain complex of finitely generated modules as an object the factor category homotopy modulo split complexes. As result are able prove that over generically Gorenstein ring is exact if and only its dual exact. This shows dependence total reflexivity conditions ring.

By defining a closure operator on effective equivalence relations in a regular category C, it is possible to establish a bijective correspondence between these closure operators and the regular epireflective subcategories of C, on the model of the closure operators on kernels in homological categories [5]. When C is an exact Goursat category [6], this correspondence restricts to a bijection bet...