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A Monadicity Theorem BULL. AUSTRAL. MATH. SOC. I8A40, I 8C I 0 , I 8C I 5 , I8D20 VOL. 17 (1977), 291-296. A monadicity theorem Francis Borceux and B.J. Day A monadicity theorem is established, for functors which satisfy the conditions of the "first isomorphism theorem" (following Linton's terminology). An application is made to the characterisation of certain types of algebraic categories generated by linear monads. Introduction In this article we establish that a functor U : 8 -»• C with a left adjoint is crudely monadic if it satisfies conditions analogous to those described by Linton [6] and there called the first isomorphism theorem. For certain types of algebraic category this monadicity theorem is an important alternative to the standard monadicity criteria of Beck [7]. In Section 2 we give a characterisation theorem based on the first isomorphism theorem for certain types of algebraic category generated by linear monads. Throughout the article we suppose, unless otherwise stated, that V = (f, ®, I, ...) is a complete and cocomplete symmetric monoidal closed category and that all categorical algebra is relative to this f . For terminology and notation we refer to Ei lenberg and Ke I ly [5] and Mac Lane VI. 1. The theorem DEFINITION 1.1. The first isomorphism theorem is said to hold for a functor U : 8 -»• C if FIT 0: 8 has coequalisers and kernel pairs, Received 2k May 1977. 29 1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 26 Sep 2021 at 23:48:02, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010480 292 Francis Borceux and B.J. Day FIT 1: f is a coequaliser in 8 iff Uf is a coequaliser in C , FIT 2: (/, g) is a kernel pair in 8 if (Uf, Ug) is a kernel pair in C . // THEOREM 1.2. If C has coequalisers and kernel pairs and F —i U : 8 ->- C , then 8 is crudely monadic over C if the first isomorphism theorem holds for U . Proof. Let T = (T, y, n) be the monad generated by the adjunction (e,n) '• F -* U : B ->• C , and let M : 8 -*• C be the canonical comparison functor where MB is UB with structure Uen : UFUB -*• UB . Because 8 has D coequalisers, M has a left adjoint M defined by the following coequaliser: FUFC FC M(C, ~-FC and the counit £ is given by: FUe B FUFUB -*• FUB * MMB FUB Applying U , we have that J/e = coequ^UFUz-, Us But Ue n FUB' MB coequaliser in C by hypothesis, so Ue is an epimorphism, so We is MB B an isomorphism. Thus e_ is a coequaliser (by hypothesis) so M reflects D isomorphisms, so U reflects isomorphisms, so e is an isomorphism. Also n : 1 •*• MM is given by Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 26 Sep 2021 at 23:48:02, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010480 A monadicity theorem 293 MFZ, MFUFC -»• MFC 'FC Let (c)> , <J> ) be the kernel pair of e in C (as in C ) . The functor U preserves kernel pairs, so Ue - coequ(U<}> , Uffo) in C by hypothesis. Thus n = coequ(^ .£/<)) , Z, .!/())„] in C . Hence it remains to show that l"|« is a monomorphism. Consider * C MM{C,t,) . Let (TT , ir_) be the kernel pair of £_ in C (as in C ) . Then the comparison 9 is a monomorphism and MJ> #6 =ir. (i = 1, 2) . But M —* M, l "i- so we have for a unique A : MP -*• Q . Thus np is a monomorphism, hence is an isomorphism. Thus [M^i .MX, M<J) .AA) is a kernel pair in C , so, by hypothesis, ((f>,-A, <(>p.A) is a kernel pair in B . Now, by definition of n, A, 6 , and a. (i = 1, 2) , the following diagram commutes U = 1, 2) : Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 26 Sep 2021 at 23:48:02, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010480 294 Francis Borceux and B.J. Day * MFC MMP But e = coequ(a , a ) in 8 because M is a reflection, and coequ(a,, a?) in C is just (C, 5) because C, is an epimorphism. Thus e = coequ(<j> Ae, <t>o^e) = coequ^A, ^A) , since e is an epimorphism. Hence A is an isomorphism because (<f » <t>5) and ('('^A, ^pA) are now both kernel pairs of e . Hence 9 is an isomorphism, so ri_ is an isomorphism. // COROLLARY 1.3. If C has eoequalisers and kernel pairs and F-<i/:8->C is an adjunction such that UF preserves aoequalisers of reflective pairs then 8 is crudely monadic over C if and only if the first isomorphism "theorem holds on U . // X 2. Linear monads Let N : A •*• C be a fully faithful dense functor. Following Diers [4], we say that an iV-theory (T, t) is algebraic if t : A •* has a right iV-adjoint R . We call an algebraic theory strictly #-hyper- linear if the mean tensor product (see [/]) C(M, C) * RtA exists in C and is flf-absolute. As usual, the category C of T-algebras is defined by the pullback: I t < Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 26 Sep 2021 at 23:48:02, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010480 A monadicity theorem 295 As a consequence (A [t, l]t[C(N-, O) = T(tA, t-) ® C{NA, C) rA ^ I C{N-, RtA) ® C(NA, C) ^ C[N-, C{NA, C) * RtA) . So we obtain F -H U where UF is just the restriction of t"-f [t, l] to C . UFC = C(NA, C) * RtA . Thus RtA & UFNA and rA ,A C(NB, UFC) c* C(NA, C) ® C(NB, RtA) S C{NA, C) ® C(NB, UFNA) . Thus the monad T = UF is strictly iV-hyperlinear in the sense of Day [3]. Such an algebraic category C will be called strongly AP-algebraic. THEOREM 2.1. If C has ooequalisers and kernel pairs and each C(NA, -) , A 6 A , preserves ooequalisers of reflective pairs, then a category 8 is strongly N-algebraio over C if and only if there exists F H U : B -»• C such that [A (1) C(M, C) ® C(NB, UFNA) 3S C(flB, t/FC) , (2) the first isomorphism theorem holds for U . Proof. Necessity follows from the fact that C(NA, -) , A € A , preserves coequalisers of reflective pairs, thus UF '=S C(NA, -) * UFA preserves coequalisers of reflective pairs, thus U creates coequalisers of reflective pairs. For sufficiency we have that U is monadic by (2) [A and Theorem 1.2. By density of N we have C(M4, C)-UFNA ^ UFC from (l). Thus the monad generated by the theory which maps A to the full image of FN (see Day [2]) coincides with the monad UF , as required. // This result should be compared with Lawvere's characterisation theorem (Linton [6], Corollary to Proposition 6) for the case C = Bru, . Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 26 Sep 2021 at 23:48:02, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010480 296 Francis Borceux and B.J. Day References [7] Francis Borceux and G.M. Ke I ly, "A notion of limit for enriched categories", Bull. Austral. Math. Soa. 12 (1975), 1*9-72. [2] B.J. Day, "Linear monads", Bull. Austral. Math. Soa. 17 (1977), 177-192. [3] B.J. Day, "On the rank of free monads" (Preprint, Department of Pure Mathematics, University of Sydney, Sydney, 1977). [4] Y. Diers, "Foncteur pleinement fidele dense classant les algebres", (Publications Internes de l'U.E.R. de Mathematiques Pures et Appliquees, 58. Universite des Sciences et Techniques de Lille I, 1975). [5] Samuel Ei lenberg and G. Max Kelly, "Closed categories", Proa. Conf. Categorical Algebra, La Jolla, California, 1965, 1*21-562 (Springer-Verlag, Berlin, Heidelberg, New York, 1966). [6] F.E.J. Linton, "Some aspects of equational categories", Proo. Conf. Categorical Algebra, La Jolla, California, 1965, 8k-9h (Springer- Verlag, Berlin, Heidelberg, New York, 1966). [7] S. Mac Lane, Categories for the working mathematician (Graduate Texts in Mathematics, 5. Springer-Verlag, Berlin, Heidelberg, New York, 1971). Institut de Mathematique pure et appliquee, Universite Catholique de Louvain, Belgi urn; Department of Pure Mathematics, University of Sydney, Sydney, New South Wales. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 26 Sep 2021 at 23:48:02, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700010480.
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